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Theorem fvco2 5493
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
StepHypRef Expression
1 fnsnfv 5481 . . . . . . 7  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 4965 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5130 . . . . . 6  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2307 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eqeq1d 2264 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( ( F  o.  G ) " { X } )  =  { x }  <->  ( F " { ( G `  X ) } )  =  { x }
) )
65abbidv 2370 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { x  |  ( ( F  o.  G
) " { X } )  =  {
x } }  =  { x  |  ( F " { ( G `
 X ) } )  =  { x } } )
76unieqd 3779 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  U. { x  |  ( ( F  o.  G ) " { X } )  =  {
x } }  =  U. { x  |  ( F " { ( G `  X ) } )  =  {
x } } )
8 df-fv 4654 . 2  |-  ( ( F  o.  G ) `
 X )  = 
U. { x  |  ( ( F  o.  G ) " { X } )  =  {
x } }
9 df-fv 4654 . 2  |-  ( F `
 ( G `  X ) )  = 
U. { x  |  ( F " {
( G `  X
) } )  =  { x } }
107, 8, 93eqtr4g 2313 1  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2242   {csn 3581   U.cuni 3768   "cima 4629    o. ccom 4630    Fn wfn 4633   ` cfv 4638
This theorem is referenced by:  fvco  5494  fvco3  5495  fvco4i  5496  ofco  5996  curry1  6109  curry2  6112  smobeth  8141  fpwwe  8201  addpqnq  8495  mulpqnq  8498  revco  11419  ccatco  11420  isoval  13594  prdsidlem  14331  gsumwmhm  14394  prdsinvlem  14530  rngidval  15270  prdsmgp  15320  lmhmco  15727  chrrhm  16412  1stccnp  17115  prdstopn  17249  xpstopnlem2  17429  uniioombllem6  18870  evlslem1  19326  evlsvar  19334  0vfval  21087  rabren3dioph  26230  dsmmbas2  26535  dsmm0cl  26538  frlmbas  26555  frlmup3  26584  frlmup4  26585  enfixsn  26589  f1lindf  26624  lindfmm  26629  f1omvdconj  26721  pmtrfinv  26734  symggen  26743  symgtrinv  26745  hausgraph  26863  stoweidlem59  27108
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 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-fv 4654
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