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Theorem funeq 5823
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 3621 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 funss 5822 . . 3 (𝐵𝐴 → (Fun 𝐴 → Fun 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4 eqimss 3620 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 funss 5822 . . 3 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
73, 6impbid 201 1 (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wss 3540  Fun wfun 5798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806
This theorem is referenced by:  funeqi  5824  funeqd  5825  fununi  5878  cnvresid  5882  fneq1  5893  funop  6320  funsndifnop  6321  nvof1o  6436  funcnvuni  7012  elpmg  7759  fundmeng  7917  isfsupp  8162  dfac9  8841  axdc3lem2  9156  frlmphllem  19938  structvtxvallem  25697  locfinreflem  29235  orvcval  29846  bnj1379  30155  bnj1385  30157  bnj1497  30382  elfunsg  31193  funop1  40327  usgredgop  40400
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