Theorem bnj519 33747

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 6600	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
3	1, 2	mpan 689	1 ⊢ (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3475  {csn 4629  ⟨cop 4635  Fun wfun 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546

This theorem is referenced by:  bnj97  33877  bnj535  33901