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Theorem fvco2 5528
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 5516 . . . . . . 7  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 5000 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 5165 . . . . . 6  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2309 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eqeq1d 2266 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( ( F  o.  G ) " { X } )  =  { x }  <->  ( F " { ( G `  X ) } )  =  { x }
) )
65abbidv 2372 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { x  |  ( ( F  o.  G
) " { X } )  =  {
x } }  =  { x  |  ( F " { ( G `
 X ) } )  =  { x } } )
76unieqd 3812 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  U. { x  |  ( ( F  o.  G ) " { X } )  =  {
x } }  =  U. { x  |  ( F " { ( G `  X ) } )  =  {
x } } )
8 df-fv 4689 . 2  |-  ( ( F  o.  G ) `
 X )  = 
U. { x  |  ( ( F  o.  G ) " { X } )  =  {
x } }
9 df-fv 4689 . 2  |-  ( F `
 ( G `  X ) )  = 
U. { x  |  ( F " {
( G `  X
) } )  =  { x } }
107, 8, 93eqtr4g 2315 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 2244   {csn 3614   U.cuni 3801   "cima 4664    o. ccom 4665    Fn wfn 4668   ` cfv 4673
This theorem is referenced by:  fvco  5529  fvco3  5530  fvco4i  5531  ofco  6031  curry1  6144  curry2  6147  smobeth  8176  fpwwe  8236  addpqnq  8530  mulpqnq  8533  revco  11454  ccatco  11455  isoval  13629  prdsidlem  14366  gsumwmhm  14429  prdsinvlem  14565  rngidval  15305  prdsmgp  15355  lmhmco  15762  chrrhm  16447  1stccnp  17150  prdstopn  17284  xpstopnlem2  17464  uniioombllem6  18905  evlslem1  19361  evlsvar  19369  0vfval  21122  rabren3dioph  26265  dsmmbas2  26570  dsmm0cl  26573  frlmbas  26590  frlmup3  26619  frlmup4  26620  enfixsn  26624  f1lindf  26659  lindfmm  26664  f1omvdconj  26756  pmtrfinv  26769  symggen  26778  symgtrinv  26780  hausgraph  26898  stoweidlem59  27143
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 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-fv 4689
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