Theorem funeq 6569

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

Assertion

Ref	Expression
funeq	⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Proof of Theorem funeq

Step	Hyp	Ref	Expression
1		eqimss2 4042	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6568	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 4041	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6568	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 211	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ⊆ wss 3949  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

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546

This theorem is referenced by:  funeqi  6570  funeqd  6571  fununi  6624  cnvresid  6628  fneq1  6641  funop  7147  funsndifnop  7149  nvof1o  7278  funcnvuni  7922  fiun  7929  elpmg  8837  fundmeng  9032  isfsupp  9365  dfac9  10131  axdc3lem2  10446  frlmphllem  21335  oldval  27349  usgredgop  28430  locfinreflem  32820  orvcval  33456  bnj1379  33841  bnj1385  33843  bnj1497  34071  funen1cnv  34091  elfunsg  34888  funop1  45991