Theorem funeq 6438

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 3974	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6437	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 3973	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6437	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 211	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ⊆ wss 3883  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-fun 6420

This theorem is referenced by:  funeqi  6439  funeqd  6440  fununi  6493  cnvresid  6497  fneq1  6508  funop  7003  funsndifnop  7005  nvof1o  7133  funcnvuni  7752  fiun  7759  elpmg  8589  fundmeng  8776  isfsupp  9062  dfac9  9823  axdc3lem2  10138  frlmphllem  20897  usgredgop  27443  locfinreflem  31692  orvcval  32324  bnj1379  32710  bnj1385  32712  bnj1497  32940  funen1cnv  32960  oldval  33965  elfunsg  34145  funop1  44662