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Theorem eqfnfv 5622
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 5568 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5568 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2295 . . 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 465 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5539 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2610 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5614 . . 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 252 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    e. cmpt 4077    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  eqfnfv2  5623  eqfnfvd  5625  eqfnfv2f  5626  fvreseq  5628  fndmdifeq0  5631  fneqeql  5633  fconst2g  5728  fnsuppres  5732  cocan1  5801  cocan2  5802  weniso  5852  tfr3  6415  ixpfi2  7154  fipreima  7161  fseqenlem1  7651  fpwwe2lem8  8259  ofsubeq0  9743  ser0f  11099  hashgval2  11360  hashf1lem1  11393  efcvgfsum  12367  prmreclem2  12964  1arithlem4  12973  1arith  12974  isgrpinv  14532  dprdf11  15258  psrbagconf1o  16120  pthaus  17332  xkohaus  17347  cnmpt11  17357  cnmpt21  17365  prdsxmetlem  17932  rolle  19337  tdeglem4  19446  resinf1o  19898  dchrelbas2  20476  dchreq  20497  nmlno0lem  21371  phoeqi  21436  occllem  21882  dfiop2  22333  hoeq  22340  ho01i  22408  hoeq1  22410  kbpj  22536  nmlnop0iALT  22575  lnopco0i  22584  nlelchi  22641  rnbra  22687  kbass5  22700  hmopidmchi  22731  hmopidmpji  22732  pjssdif2i  22754  pjinvari  22771  subfacp1lem3  23713  subfacp1lem5  23715  fprb  24129  rdgprc  24151  eqeefv  24531  axlowdimlem14  24583  surjsec2  25120  cocanfo  26374  eqfnun  26387  sdclem2  26452  rrnmet  26553  rrnequiv  26559  fnnfpeq0  26758  pw2f1ocnv  27130  islindf4  27308  caofcan  27540  addrcom  27680  bnj1542  28889  bnj580  28945  ltrnid  30324  ltrneq2  30337  tendoeq1  30953
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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