MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqfunfv Structured version   Unicode version

Theorem eqfunfv 5825
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 5475 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 funfn 5475 . 2  |-  ( Fun 
G  <->  G  Fn  dom  G )
3 eqfnfv2 5821 . 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 1652   A.wral 2698   dom cdm 4871   Fun wfun 5441    Fn wfn 5442   ` cfv 5447
This theorem is referenced by:  fnpr  5943  fnprOLD  5944  nodenselem5  25633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-fv 5455
  Copyright terms: Public domain W3C validator