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Theorem funeq 5101
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 3118 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 funss 5100 . . 3 (𝐵𝐴 → (Fun 𝐴 → Fun 𝐵))
31, 2syl 14 . 2 (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4 eqimss 3117 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 funss 5100 . . 3 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
64, 5syl 14 . 2 (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
73, 6impbid 128 1 (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wss 3037  Fun wfun 5075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-in 3043  df-ss 3050  df-br 3896  df-opab 3950  df-rel 4506  df-cnv 4507  df-co 4508  df-fun 5083
This theorem is referenced by:  funeqi  5102  funeqd  5103  fununi  5149  funcnvuni  5150  cnvresid  5155  fneq1  5169  elpmg  6512  fundmeng  6655
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