Theorem bnj519 32694

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

Step	Hyp	Ref	Expression
1		bnj519.1	. 2 ⊢ 𝐴 ∈ V
2		funsng 6481	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
3	1, 2	mpan 686	1 ⊢ (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3430  {csn 4566  ⟨cop 4572  Fun wfun 6424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-fun 6432

This theorem is referenced by:  bnj97  32825  bnj535  32849