Theorem funeq 5337

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 3279	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 5336	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 14	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 3278	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 5336	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 14	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 129	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395   ⊆ wss 3197  Fun wfun 5311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210  df-br 4083  df-opab 4145  df-rel 4725  df-cnv 4726  df-co 4727  df-fun 5319

This theorem is referenced by:  funeqi  5338  funeqd  5339  fununi  5388  funcnvuni  5389  cnvresid  5394  fneq1  5408  funop  5817  elpmg  6809  fundmeng  6958  usgredgop  15965