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Theorem fvco2 5594
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 5582 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 5012 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5178 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2334 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2350 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 5240 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3 5521 . 2  |-  ( ( F  o.  G ) `
 X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )
8 dffv3 5521 . 2  |-  ( F `
 ( G `  X ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) )
96, 7, 83eqtr4g 2340 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 358    = wceq 1623    e. wcel 1684   {csn 3640   "cima 4692    o. ccom 4693   iotacio 5217    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  fvco  5595  fvco3  5596  fvco4i  5597  ofco  6097  curry1  6210  curry2  6213  smobeth  8208  fpwwe  8268  addpqnq  8562  mulpqnq  8565  revco  11489  ccatco  11490  isoval  13667  prdsidlem  14404  gsumwmhm  14467  prdsinvlem  14603  rngidval  15343  prdsmgp  15393  lmhmco  15800  chrrhm  16485  1stccnp  17188  prdstopn  17322  xpstopnlem2  17502  uniioombllem6  18943  evlslem1  19399  evlsvar  19407  0vfval  21162  cnre2csqlem  23294  rabren3dioph  26898  dsmmbas2  27203  dsmm0cl  27206  frlmbas  27223  frlmup3  27252  frlmup4  27253  enfixsn  27257  f1lindf  27292  lindfmm  27297  f1omvdconj  27389  pmtrfinv  27402  symggen  27411  symgtrinv  27413  hausgraph  27531  stoweidlem59  27808  afvco2  28037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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