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Theorem funeq 5877
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
StepHypRef Expression
1 eqimss2 3643 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 funss 5876 . . 3 (𝐵𝐴 → (Fun 𝐴 → Fun 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4 eqimss 3642 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 funss 5876 . . 3 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
73, 6impbid 202 1 (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wss 3560  Fun wfun 5851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-in 3567  df-ss 3574  df-br 4624  df-opab 4684  df-rel 5091  df-cnv 5092  df-co 5093  df-fun 5859
This theorem is referenced by:  funeqi  5878  funeqd  5879  fununi  5932  cnvresid  5936  fneq1  5947  funop  6379  funsndifnop  6381  nvof1o  6501  funcnvuni  7081  elpmg  7833  fundmeng  7991  isfsupp  8239  dfac9  8918  axdc3lem2  9233  frlmphllem  20059  usgredgop  25992  locfinreflem  29731  orvcval  30342  bnj1379  30662  bnj1385  30664  bnj1497  30889  elfunsg  31718  funop1  40629
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