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Theorem funeq 5274
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 3231 . . 3  |-  ( A  =  B  ->  B  C_  A )
2 funss 5273 . . 3  |-  ( B 
C_  A  ->  ( Fun  A  ->  Fun  B ) )
31, 2syl 15 . 2  |-  ( A  =  B  ->  ( Fun  A  ->  Fun  B ) )
4 eqimss 3230 . . 3  |-  ( A  =  B  ->  A  C_  B )
5 funss 5273 . . 3  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
64, 5syl 15 . 2  |-  ( A  =  B  ->  ( Fun  B  ->  Fun  A ) )
73, 6impbid 183 1  |-  ( A  =  B  ->  ( Fun  A  <->  Fun  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    C_ wss 3152   Fun wfun 5249
This theorem is referenced by:  funeqi  5275  funeqd  5276  fununi  5316  funcnvuni  5317  cnvresid  5322  fneq1  5333  elpmg  6786  fundmeng  6935  dfac9  7762  axdc3lem2  8077  nvof1o  23036  measbasedom  23532  orvcval  23658  elfunsg  24455  injrec2  25119  cur1val  25198  isalg  25721  algi  25727  tartarmap  25888  pfsubkl  26047  bnj1379  28863  bnj1385  28865  bnj1497  29090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-fun 5257
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