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Theorem fvco2 5446
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 5434 . . . . . . 7  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 4919 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5084 . . . . . 6  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2304 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eqeq1d 2261 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( ( F  o.  G ) " { X } )  =  { x }  <->  ( F " { ( G `  X ) } )  =  { x }
) )
65abbidv 2363 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { x  |  ( ( F  o.  G
) " { X } )  =  {
x } }  =  { x  |  ( F " { ( G `
 X ) } )  =  { x } } )
76unieqd 3738 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  U. { x  |  ( ( F  o.  G ) " { X } )  =  {
x } }  =  U. { x  |  ( F " { ( G `  X ) } )  =  {
x } } )
8 df-fv 4608 . 2  |-  ( ( F  o.  G ) `
 X )  = 
U. { x  |  ( ( F  o.  G ) " { X } )  =  {
x } }
9 df-fv 4608 . 2  |-  ( F `
 ( G `  X ) )  = 
U. { x  |  ( F " {
( G `  X
) } )  =  { x } }
107, 8, 93eqtr4g 2310 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 2239   {csn 3544   U.cuni 3727   "cima 4583    o. ccom 4584    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  fvco  5447  fvco3  5448  fvco4i  5449  ofco  5949  curry1  6062  curry2  6065  smobeth  8088  fpwwe  8148  addpqnq  8442  mulpqnq  8445  revco  11366  ccatco  11367  isoval  13511  prdsidlem  14239  gsumwmhm  14302  prdsinvlem  14438  rngidval  15178  prdsmgp  15228  lmhmco  15635  chrrhm  16317  1stccnp  17020  prdstopn  17154  xpstopnlem2  17334  uniioombllem6  18775  evlslem1  19231  evlsvar  19239  0vfval  20992  rabren3dioph  26064  dsmmbas2  26369  dsmm0cl  26372  frlmbas  26389  frlmup3  26418  frlmup4  26419  enfixsn  26423  f1lindf  26458  lindfmm  26463  f1omvdconj  26555  pmtrfinv  26568  symggen  26577  symgtrinv  26579  hausgraph  26697
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608
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