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Theorem eqfunfv 5764
Description: Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
Assertion
Ref Expression
eqfunfv  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G  /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
Distinct variable groups:    x, F    x, G

Proof of Theorem eqfunfv
StepHypRef Expression
1 funfn 5415 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 funfn 5415 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
3 eqfnfv2 5760 . 2  |-  ( ( F  Fn  dom  F  /\  G  Fn  dom  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G  /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
41, 2, 3syl2anb 466 1  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  =  G  <->  ( dom  F  =  dom  G  /\  A. x  e.  dom  F ( F `  x )  =  ( G `  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   A.wral 2642   dom cdm 4811   Fun wfun 5381    Fn wfn 5382   ` cfv 5387
This theorem is referenced by:  fnpr  5882  fnprOLD  5883  nodenselem5  25356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-fv 5395
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