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Theorem funeq 5099
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 3116 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5098 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 14 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3115 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5098 . . 3  |-  ( A 
C_  B  ->  ( 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 1312    C_ wss 3035   Fun wfun 5073
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 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-in 3041  df-ss 3048  df-br 3894  df-opab 3948  df-rel 4504  df-cnv 4505  df-co 4506  df-fun 5081
This theorem is referenced by:  funeqi  5100  funeqd  5101  fununi  5147  funcnvuni  5148  cnvresid  5153  fneq1  5167  elpmg  6510  fundmeng  6653
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