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Theorem eqfnfv 5982
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 5918 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5918 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2476 . . 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 479 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5882 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2818 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5971 . . 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 5 . 2  |-  ( ( x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `
 x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
94, 8syl6bb 261 1  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    |-> cmpt 4515    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  eqfnfv2  5983  eqfnfvd  5985  eqfnfv2f  5986  fvreseq0  5988  fnmptfvd  5991  fndmdifeq0  5994  fneqeql  5996  fnnfpeq0  6103  fconst2g  6127  fnsuppresOLD  6132  cocan1  6195  cocan2  6196  weniso  6251  fnsuppres  6945  tfr3  7086  ixpfi2  7836  fipreima  7844  fseqenlem1  8422  fpwwe2lem8  9032  ofsubeq0  10553  ser0f  12163  hashgval2  12449  hashf1lem1  12508  prodf1f  13713  efcvgfsum  13833  prmreclem2  14447  1arithlem4  14456  1arith  14457  isgrpinv  16227  dprdf11  17190  dprdf11OLD  17197  psrbagconf1o  18153  islindf4  19000  pthaus  20265  xkohaus  20280  cnmpt11  20290  cnmpt21  20298  prdsxmetlem  20997  rrxmet  21961  rolle  22517  tdeglem4  22584  resinf1o  23049  dchrelbas2  23638  dchreq  23659  eqeefv  24333  axlowdimlem14  24385  nmlno0lem  25835  phoeqi  25900  occllem  26348  dfiop2  26799  hoeq  26806  ho01i  26874  hoeq1  26876  kbpj  27002  nmlnop0iALT  27041  lnopco0i  27050  nlelchi  27107  rnbra  27153  kbass5  27166  hmopidmchi  27197  hmopidmpji  27198  pjssdif2i  27220  pjinvari  27237  subfacp1lem3  28823  subfacp1lem5  28825  mrsubff1  29071  msubff1  29113  faclimlem1  29386  fprb  29420  rdgprc  29444  cocanfo  30413  eqfnun  30417  sdclem2  30440  rrnmet  30530  rrnequiv  30536  pw2f1ocnv  31183  caofcan  31432  addrcom  31588  dvnprodlem1  31946  bnj1542  34058  bnj580  34114  ltrnid  36002  ltrneq2  36015  tendoeq1  36633
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