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

Theorem eqfnfv 5556
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 5502 . . 3  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
2 dffn5 5502 . . 3  |-  ( G  Fn  A  <->  G  =  ( x  e.  A  |->  ( G `  x
) ) )
3 eqeq12 2270 . . 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 467 . 2  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( x  e.  A  |->  ( F `  x
) )  =  ( x  e.  A  |->  ( G `  x ) ) ) )
5 fvex 5472 . . . 4  |-  ( F `
 x )  e. 
_V
65rgenw 2585 . . 3  |-  A. x  e.  A  ( F `  x )  e.  _V
7 mpteqb 5548 . . 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 10 . 2  |-  ( ( x  e.  A  |->  ( F `  x ) )  =  ( x  e.  A  |->  ( G `
 x ) )  <->  A. x  e.  A  ( F `  x )  =  ( G `  x ) )
94, 8syl6bb 254 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   _Vcvv 2763    e. cmpt 4051    Fn wfn 4668   ` cfv 4673
This theorem is referenced by:  eqfnfv2  5557  eqfnfvd  5559  eqfnfv2f  5560  fvreseq  5562  fndmdifeq0  5565  fneqeql  5567  fconst2g  5662  fnsuppres  5666  cocan1  5735  cocan2  5736  weniso  5786  tfr3  6383  ixpfi2  7122  fipreima  7129  fseqenlem1  7619  fpwwe2lem8  8227  ofsubeq0  9711  ser0f  11065  hashgval2  11326  hashf1lem1  11358  efcvgfsum  12329  prmreclem2  12926  1arithlem4  12935  1arith  12936  isgrpinv  14494  dprdf11  15220  psrbagconf1o  16082  pthaus  17294  xkohaus  17309  cnmpt11  17319  cnmpt21  17327  prdsxmetlem  17894  rolle  19299  tdeglem4  19408  resinf1o  19860  dchrelbas2  20438  dchreq  20459  nmlno0lem  21331  phoeqi  21396  occllem  21842  dfiop2  22293  hoeq  22300  ho01i  22368  hoeq1  22370  kbpj  22496  nmlnop0iALT  22535  lnopco0i  22544  nlelchi  22601  rnbra  22647  kbass5  22660  hmopidmchi  22691  hmopidmpji  22692  pjssdif2i  22714  pjinvari  22731  subfacp1lem3  23085  subfacp1lem5  23087  fprb  23498  rdgprc  23520  eqeefv  23906  axlowdimlem14  23958  surjsec2  24487  repfuntw  24527  cocanfo  25741  eqfnun  25754  sdclem2  25819  rrnmet  25920  rrnequiv  25926  fnnfpeq0  26125  pw2f1ocnv  26497  islindf4  26675  caofcan  26907  addrcom  27048  bnj1542  27938  bnj580  27994  ltrnid  29491  ltrneq2  29504  tendoeq1  30120
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-fv 4689
  Copyright terms: Public domain W3C validator