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Theorem eqfnfv 5827
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
eqfnfv  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
Distinct variable groups:    x, A    x, F    x, G

Proof of Theorem eqfnfv
StepHypRef Expression
1 dffn5 5772 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5772 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2448 . . 3  |-  ( ( F  =  ( x  e.  A  |->  ( F `
 x ) )  /\  G  =  ( x  e.  A  |->  ( G `  x ) ) )  ->  ( F  =  G  <->  ( x  e.  A  |->  ( F `
 x ) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
41, 2, 3syl2anb 466 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5742 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2773 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5819 . . 3  |-  ( A. x  e.  A  ( F `  x )  e.  _V  ->  ( (
x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `
 x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
86, 7ax-mp 8 . 2  |-  ( ( x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `
 x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
94, 8syl6bb 253 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    e. cmpt 4266    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  eqfnfv2  5828  eqfnfvd  5830  eqfnfv2f  5831  fvreseq  5833  fndmdifeq0  5836  fneqeql  5838  fconst2g  5946  fnsuppres  5952  cocan1  6024  cocan2  6025  weniso  6075  tfr3  6660  ixpfi2  7405  fipreima  7412  fseqenlem1  7905  fpwwe2lem8  8512  ofsubeq0  9997  ser0f  11376  hashgval2  11652  hashf1lem1  11704  efcvgfsum  12688  prmreclem2  13285  1arithlem4  13294  1arith  13295  isgrpinv  14855  dprdf11  15581  psrbagconf1o  16439  pthaus  17670  xkohaus  17685  cnmpt11  17695  cnmpt21  17703  prdsxmetlem  18398  rolle  19874  tdeglem4  19983  resinf1o  20438  dchrelbas2  21021  dchreq  21042  nmlno0lem  22294  phoeqi  22359  occllem  22805  dfiop2  23256  hoeq  23263  ho01i  23331  hoeq1  23333  kbpj  23459  nmlnop0iALT  23498  lnopco0i  23507  nlelchi  23564  rnbra  23610  kbass5  23623  hmopidmchi  23654  hmopidmpji  23655  pjssdif2i  23677  pjinvari  23694  subfacp1lem3  24868  subfacp1lem5  24870  prodf1f  25220  faclimlem1  25362  fprb  25397  rdgprc  25422  eqeefv  25842  axlowdimlem14  25894  cocanfo  26419  eqfnun  26423  sdclem2  26446  rrnmet  26538  rrnequiv  26544  fnnfpeq0  26739  pw2f1ocnv  27108  islindf4  27285  caofcan  27517  addrcom  27656  bnj1542  29228  bnj580  29284  ltrnid  30932  ltrneq2  30945  tendoeq1  31561
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 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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