Theorem funeq 6587

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 4054	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6586	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 4053	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6586	. . 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 1536   ⊆ wss 3962  Fun wfun 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ss 3979  df-br 5148  df-opab 5210  df-rel 5695  df-cnv 5696  df-co 5697  df-fun 6564

This theorem is referenced by:  funeqi  6588  funeqd  6589  fununi  6642  cnvresid  6646  fneq1  6659  funop  7168  funsndifnop  7170  nvof1o  7299  funcnvuni  7954  fiun  7965  elpmg  8881  fundmeng  9070  isfsupp  9402  dfac9  10174  axdc3lem2  10488  frlmphllem  21817  psdmul  22187  oldval  27907  usgredgop  29201  locfinreflem  33800  orvcval  34438  bnj1379  34822  bnj1385  34824  bnj1497  35052  funen1cnv  35080  elfunsg  35897  modelaxreplem1  44942  modelaxreplem2  44943  modelaxrep  44945  funop1  47232