Theorem funeq 5908

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 3658	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 5907	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 3657	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 5907	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 202	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ⊆ wss 3574  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  funeqi  5909  funeqd  5910  fununi  5964  cnvresid  5968  fneq1  5979  funop  6414  funsndifnop  6416  nvof1o  6536  funcnvuni  7119  elpmg  7873  fundmeng  8031  isfsupp  8279  dfac9  8958  axdc3lem2  9273  frlmphllem  20119  usgredgop  26065  locfinreflem  29907  orvcval  30519  bnj1379  30901  bnj1385  30903  bnj1497  31128  elfunsg  32023  funop1  41302