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Theorem fvco2 5610
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 5598 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 5028 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5194 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2347 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2363 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 5256 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3 5537 . 2  |-  ( ( F  o.  G ) `
 X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )
8 dffv3 5537 . 2  |-  ( F `
 ( G `  X ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) )
96, 7, 83eqtr4g 2353 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 1632    e. wcel 1696   {csn 3653   "cima 4708    o. ccom 4709   iotacio 5233    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  fvco  5611  fvco3  5612  fvco4i  5613  ofco  6113  curry1  6226  curry2  6229  smobeth  8224  fpwwe  8284  addpqnq  8578  mulpqnq  8581  revco  11505  ccatco  11506  isoval  13683  prdsidlem  14420  gsumwmhm  14483  prdsinvlem  14619  rngidval  15359  prdsmgp  15409  lmhmco  15816  chrrhm  16501  1stccnp  17204  prdstopn  17338  xpstopnlem2  17518  uniioombllem6  18959  evlslem1  19415  evlsvar  19423  0vfval  21178  cnre2csqlem  23309  rabren3dioph  27001  dsmmbas2  27306  dsmm0cl  27309  frlmbas  27326  frlmup3  27355  frlmup4  27356  enfixsn  27360  f1lindf  27395  lindfmm  27400  f1omvdconj  27492  pmtrfinv  27505  symggen  27514  symgtrinv  27516  hausgraph  27634  stoweidlem59  27911  afvco2  28144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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