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Theorem fvco2 5757
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 5745 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 5162 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5334 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2455 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2471 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 5398 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3 5683 . 2  |-  ( ( F  o.  G ) `
 X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )
8 dffv3 5683 . 2  |-  ( F `
 ( G `  X ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) )
96, 7, 83eqtr4g 2461 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 1649    e. wcel 1721   {csn 3774   "cima 4840    o. ccom 4841   iotacio 5375    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  fvco  5758  fvco3  5759  fvco4i  5760  ofco  6283  curry1  6397  curry2  6400  smobeth  8417  fpwwe  8477  addpqnq  8771  mulpqnq  8774  revco  11758  ccatco  11759  isoval  13945  prdsidlem  14682  gsumwmhm  14745  prdsinvlem  14881  rngidval  15621  prdsmgp  15671  lmhmco  16074  chrrhm  16767  1stccnp  17478  prdstopn  17613  xpstopnlem2  17796  uniioombllem6  19433  evlslem1  19889  evlsvar  19897  0vfval  22038  cnre2csqlem  24261  mblfinlem  26143  rabren3dioph  26766  dsmmbas2  27071  dsmm0cl  27074  frlmbas  27091  frlmup3  27120  frlmup4  27121  enfixsn  27125  f1lindf  27160  lindfmm  27165  f1omvdconj  27257  pmtrfinv  27270  symggen  27279  symgtrinv  27281  hausgraph  27399  stoweidlem59  27675  afvco2  27907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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