Theorem funALTVeq 39718

Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
funALTVeq	⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

Proof of Theorem funALTVeq

Step	Hyp	Ref	Expression
1		eqimss2 3967	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funALTVss 39717	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ( FunALTV 𝐴 → FunALTV 𝐵))
3	1, 2	syl 18	. 2 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 → FunALTV 𝐵))
4		eqimss 3966	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funALTVss 39717	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
6	4, 5	syl 18	. 2 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
7	3, 6	impbid 215	1 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ⊆ wss 3895   FunALTV wfunALTV 39149

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-11 2193  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-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-dm 5660  df-rn 5661  df-res 5662  df-coss 39434  df-cnvrefrel 39540  df-funALTV 39700

This theorem is used by:  funALTVeqi  39719  funALTVeqd  39720