Theorem funeq 6556

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 4018	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6555	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 4017	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6555	. . 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 3926  Fun wfun 6525

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ss 3943  df-br 5120  df-opab 5182  df-rel 5661  df-cnv 5662  df-co 5663  df-fun 6533

This theorem is referenced by:  funeqi  6557  funeqd  6558  fununi  6611  cnvresid  6615  fneq1  6629  funop  7139  funsndifnop  7141  nvof1o  7273  funcnvuni  7928  fiun  7941  elpmg  8857  fundmeng  9046  isfsupp  9377  dfac9  10151  axdc3lem2  10465  frlmphllem  21740  psdmul  22104  oldval  27814  usgredgop  29149  locfinreflem  33871  orvcval  34490  bnj1379  34861  bnj1385  34863  bnj1497  35091  funen1cnv  35119  elfunsg  35934  modelaxreplem1  45003  modelaxreplem2  45004  modelaxrep  45006  funop1  47312