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 3995	. . 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 3994	. . 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 1542   ⊆ wss 3903  Fun wfun 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-fun 6502

This theorem is referenced by:  funeqi  6521  funeqd  6522  fununi  6575  cnvresid  6579  fneq1  6591  funop  7104  funsndifnop  7106  nvof1o  7236  funcnvuni  7884  fiun  7897  elpmg  8792  fundmeng  8981  isfsupp  9280  dfac9  10059  axdc3lem2  10373  frlmphllem  21747  psdmul  22121  oldval  27842  usgredgop  29255  locfinreflem  34018  orvcval  34636  bnj1379  35006  bnj1385  35008  bnj1497  35236  funen1cnv  35265  elfunsg  36130  modelaxreplem1  45334  modelaxreplem2  45335  modelaxrep  45337  funop1  47643