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Theorem eqfnfv 5975
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 5913 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5913 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2486 . . 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 5876 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2825 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5964 . . 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 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    |-> cmpt 4505    Fn wfn 5583   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by:  eqfnfv2  5976  eqfnfvd  5978  eqfnfv2f  5979  fvreseq0  5981  fnmptfvd  5984  fndmdifeq0  5987  fneqeql  5989  fnnfpeq0  6092  fconst2g  6115  fnsuppresOLD  6121  cocan1  6182  cocan2  6183  weniso  6238  fnsuppres  6927  tfr3  7068  ixpfi2  7818  fipreima  7826  fseqenlem1  8405  fpwwe2lem8  9015  ofsubeq0  10533  ser0f  12128  hashgval2  12414  hashf1lem1  12470  efcvgfsum  13683  prmreclem2  14294  1arithlem4  14303  1arith  14304  isgrpinv  15910  dprdf11  16865  dprdf11OLD  16872  psrbagconf1o  17825  islindf4  18668  pthaus  19902  xkohaus  19917  cnmpt11  19927  cnmpt21  19935  prdsxmetlem  20634  rrxmet  21598  rolle  22154  tdeglem4  22221  resinf1o  22684  dchrelbas2  23268  dchreq  23289  eqeefv  23910  axlowdimlem14  23962  nmlno0lem  25412  phoeqi  25477  occllem  25925  dfiop2  26376  hoeq  26383  ho01i  26451  hoeq1  26453  kbpj  26579  nmlnop0iALT  26618  lnopco0i  26627  nlelchi  26684  rnbra  26730  kbass5  26743  hmopidmchi  26774  hmopidmpji  26775  pjssdif2i  26797  pjinvari  26814  signstres  28200  subfacp1lem3  28294  subfacp1lem5  28296  prodf1f  28631  faclimlem1  28773  fprb  28808  rdgprc  28832  cocanfo  29839  eqfnun  29843  sdclem2  29866  rrnmet  29956  rrnequiv  29962  pw2f1ocnv  30611  caofcan  30856  addrcom  30990  bnj1542  33012  bnj580  33068  ltrnid  34949  ltrneq2  34962  tendoeq1  35578
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