Theorem funeq 6502

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 3995	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6501	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 3994	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6501	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 212	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ⊆ wss 3903  Fun wfun 6476

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ss 3920  df-br 5093  df-opab 5155  df-rel 5626  df-cnv 5627  df-co 5628  df-fun 6484

This theorem is referenced by:  funeqi  6503  funeqd  6504  fununi  6557  cnvresid  6561  fneq1  6573  funop  7083  funsndifnop  7085  nvof1o  7217  funcnvuni  7865  fiun  7878  elpmg  8770  fundmeng  8957  isfsupp  9255  dfac9  10031  axdc3lem2  10345  frlmphllem  21687  psdmul  22051  oldval  27764  usgredgop  29115  locfinreflem  33807  orvcval  34426  bnj1379  34797  bnj1385  34799  bnj1497  35027  funen1cnv  35055  elfunsg  35894  modelaxreplem1  44956  modelaxreplem2  44957  modelaxrep  44959  funop1  47271