Theorem funALTVeq 39462

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 3995	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funALTVss 39461	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ( FunALTV 𝐴 → FunALTV 𝐵))
3	1, 2	syl 18	. 2 ⊢ (𝐴 = 𝐵 → ( FunALTV 𝐴 → FunALTV 𝐵))
4		eqimss 3994	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funALTVss 39461	. . 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 3904   FunALTV wfunALTV 38893

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-coss 39178  df-cnvrefrel 39284  df-funALTV 39444

This theorem is used by:  funALTVeqi  39463  funALTVeqd  39464