Theorem funeq 6378

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 4027	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6377	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 4026	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6377	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 214	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ⊆ wss 3939  Fun wfun 6352

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-in 3946  df-ss 3955  df-br 5070  df-opab 5132  df-rel 5565  df-cnv 5566  df-co 5567  df-fun 6360

This theorem is referenced by:  funeqi  6379  funeqd  6380  fununi  6432  cnvresid  6436  fneq1  6447  funop  6914  funsndifnop  6916  nvof1o  7040  funcnvuni  7639  fiun  7647  elpmg  8425  fundmeng  8587  isfsupp  8840  dfac9  9565  axdc3lem2  9876  frlmphllem  20927  usgredgop  26958  locfinreflem  31108  orvcval  31719  bnj1379  32106  bnj1385  32108  bnj1497  32336  funen1cnv  32361  elfunsg  33381  funop1  43489