Theorem funeq 6520

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 4003	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6519	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 4002	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6519	. . 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 3911  Fun wfun 6493

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 3928  df-br 5103  df-opab 5165  df-rel 5638  df-cnv 5639  df-co 5640  df-fun 6501

This theorem is referenced by:  funeqi  6521  funeqd  6522  fununi  6575  cnvresid  6579  fneq1  6591  funop  7103  funsndifnop  7105  nvof1o  7237  funcnvuni  7888  fiun  7901  elpmg  8793  fundmeng  8980  isfsupp  9292  dfac9  10066  axdc3lem2  10380  frlmphllem  21722  psdmul  22086  oldval  27799  usgredgop  29150  locfinreflem  33823  orvcval  34442  bnj1379  34813  bnj1385  34815  bnj1497  35043  funen1cnv  35071  elfunsg  35897  modelaxreplem1  44961  modelaxreplem2  44962  modelaxrep  44964  funop1  47277