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

Theorem eqfnfv 5998
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 5932 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5932 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2474 . . 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 486 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5897 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2760 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5986 . . 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 269 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 189    /\ wa 375    = wceq 1454    e. wcel 1897   A.wral 2748   _Vcvv 3056    |-> cmpt 4474    Fn wfn 5595   ` cfv 5600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-fv 5608
This theorem is referenced by:  eqfnfv2  5999  eqfnfvd  6001  eqfnfv2f  6002  fvreseq0  6004  fnmptfvd  6007  fndmdifeq0  6010  fneqeql  6012  fnnfpeq0  6118  fconst2g  6142  cocan1  6213  cocan2  6214  weniso  6269  fnsuppres  6968  tfr3  7142  ixpfi2  7897  fipreima  7905  fseqenlem1  8480  fpwwe2lem8  9087  ofsubeq0  10633  ser0f  12297  hashgval2  12588  hashf1lem1  12650  prodf1f  13996  efcvgfsum  14188  prmreclem2  14909  1arithlem4  14918  1arith  14919  isgrpinv  16764  dprdf11  17704  psrbagconf1o  18646  islindf4  19444  pthaus  20701  xkohaus  20716  cnmpt11  20726  cnmpt21  20734  prdsxmetlem  21431  rrxmet  22410  rolle  22990  tdeglem4  23057  resinf1o  23533  dchrelbas2  24213  dchreq  24234  eqeefv  24981  axlowdimlem14  25033  nmlno0lem  26482  phoeqi  26547  occllem  27004  dfiop2  27454  hoeq  27461  ho01i  27529  hoeq1  27531  kbpj  27657  nmlnop0iALT  27696  lnopco0i  27705  nlelchi  27762  rnbra  27808  kbass5  27821  hmopidmchi  27852  hmopidmpji  27853  pjssdif2i  27875  pjinvari  27892  bnj1542  29716  bnj580  29772  subfacp1lem3  29953  subfacp1lem5  29955  mrsubff1  30200  msubff1  30242  faclimlem1  30427  fprb  30461  rdgprc  30489  broucube  32018  cocanfo  32088  eqfnun  32092  sdclem2  32115  rrnmet  32205  rrnequiv  32211  ltrnid  33744  ltrneq2  33757  tendoeq1  34375  pw2f1ocnv  35936  caofcan  36715  addrcom  36871  dvnprodlem1  37858
  Copyright terms: Public domain W3C validator