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Theorem bnj519 30058
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj519.1 𝐴 ∈ V
Assertion
Ref Expression
bnj519 (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})

Proof of Theorem bnj519
StepHypRef Expression
1 bnj519.1 . 2 𝐴 ∈ V
2 funsng 5851 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
31, 2mpan 702 1 (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by:  bnj97  30190  bnj535  30214
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