Theorem bnj519 35367

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 6591	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
3	1, 2	mpan 703	1 ⊢ (𝐵 ∈ V → Fun {⟨𝐴, 𝐵⟩})

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  Vcvv 3450  {csn 4583  ⟨cop 4589  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-fun 6540

This theorem is used by:  bnj97  35496  bnj535  35520