http://us2.metamath.org:8888/mpeuni/mmrecent.html Recent proofs for Metamath proof system en http://us2.metamath.org:8888/mpeuni/dfmpt2.html 29-Aug-2010df-mpt2 4179 (although it requires that C be a set).
   ⊢ C ∈ V    ⇒
   ⊢ (x ∈ A, y ∈ B ↦ C) = ∪x ∈ A ∪y ∈ B {x, y, C}
    ]]> http://us2.metamath.org:8888/mpeuni/dfmpt.html 24-Aug-2010df-mpt 4178 (although it requires that B be a set).
   ⊢ B ∈ V    ⇒
   ⊢ (x ∈ A ↦ B) = ∪x ∈ A {x, B}
    ]]> http://us2.metamath.org:8888/mpeuni/nfunsn.html 9-Aug-2010   ⊢ (¬ Fun (F ↾ {A}) → (F ‘A) = ∅)
    ]]> http://us2.metamath.org:8888/mpeuni/dffv2.html 7-Aug-2010df-fv 3319 that doesn't require dummy variables.
   ⊢ (F ‘A) = ∪((F “ {A}) ∖ ∪∪(((F ↾ {A}) ∘ ^◡(F ↾ {A})) ∖ I))
    ]]> http://us2.metamath.org:8888/mpeuni/elimasni.html 5-Aug-2010   ⊢ (C ∈ (A “ {B}) → BAC)
    ]]> http://us2.metamath.org:8888/mpeuni/dfdm2.html 3-Aug-2010df-dm 3309 that doesn't require dummy variables.
   ⊢ dom A = ∪∪(^◡A ∘ A)
    ]]> http://us2.metamath.org:8888/mpeuni/uniintsn.html 2-Aug-2010A is a singleton." See also en1 4613 and card1 5027.
   ⊢ (∪A = ∩A ↔ ∃x A = {x})
    ]]> http://us2.metamath.org:8888/mpeuni/pwjust.html 12-Jul-2010df-pw 2497. (Contributed by Rodolfo Medina, 28-Apr-2010.)
   ⊢ {x∣x ⊆ A} = {y∣y ⊆ A}
    ]]> http://us2.metamath.org:8888/mpeuni/sbss.html 12-Jul-2010   ⊢ ([y / x]x ⊆ A ↔ y ⊆ A)
    ]]> http://us2.metamath.org:8888/mpeuni/sbsslem.html 12-Jul-2010sbss 2451.
   ⊢ ([y / x]x ⊆ A ↔ ∀w((w ∈ y ⋀ w ∈ A) ↔ w ∈ y))]]>