Theorem bnj519 32760

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 6514	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
3	1, 2	mpan 688	1 ⊢ (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})

Syntax hints:   → wi 4   ∈ wcel 2104  Vcvv 3437  {csn 4565  ⟨cop 4571  Fun wfun 6452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-fun 6460

This theorem is referenced by:  bnj97  32891  bnj535  32915