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Theorem funeq 5035
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 3079 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5034 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 14 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3078 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5034 . . 3  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
64, 5syl 14 . 2  |-  ( A  =  B  ->  ( Fun  B  ->  Fun  A ) )
73, 6impbid 127 1  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    C_ wss 2999   Fun wfun 5009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-in 3005  df-ss 3012  df-br 3846  df-opab 3900  df-rel 4445  df-cnv 4446  df-co 4447  df-fun 5017
This theorem is referenced by:  funeqi  5036  funeqd  5037  fununi  5082  funcnvuni  5083  cnvresid  5088  fneq1  5102  elpmg  6419  fundmeng  6522
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