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Theorem fvco2 5789
Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
fvco2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )

Proof of Theorem fvco2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fnsnfv 5777 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 5194 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5366 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2486 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2502 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 5430 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3 5715 . 2  |-  ( ( F  o.  G ) `
 X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )
8 dffv3 5715 . 2  |-  ( F `
 ( G `  X ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) )
96, 7, 83eqtr4g 2492 1  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3806   "cima 4872    o. ccom 4873   iotacio 5407    Fn wfn 5440   ` cfv 5445
This theorem is referenced by:  fvco  5790  fvco3  5791  fvco4i  5792  ofco  6315  curry1  6429  curry2  6432  smobeth  8450  fpwwe  8510  addpqnq  8804  mulpqnq  8807  revco  11791  ccatco  11792  isoval  13978  prdsidlem  14715  gsumwmhm  14778  prdsinvlem  14914  rngidval  15654  prdsmgp  15704  lmhmco  16107  chrrhm  16800  1stccnp  17513  prdstopn  17648  xpstopnlem2  17831  uniioombllem6  19468  evlslem1  19924  evlsvar  19932  0vfval  22073  cnre2csqlem  24296  mblfinlem  26190  rabren3dioph  26813  dsmmbas2  27118  dsmm0cl  27121  frlmbas  27138  frlmup3  27167  frlmup4  27168  enfixsn  27172  f1lindf  27207  lindfmm  27212  f1omvdconj  27304  pmtrfinv  27317  symggen  27326  symgtrinv  27328  hausgraph  27446  stoweidlem59  27722  afvco2  27954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-fv 5453
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