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Theorem fvco2 5606
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 imaco 5152 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
2 fnsnfv 5596 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
32imaeq2d 4988 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
41, 3eqtr4id 2241 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2259 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 5218 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3g 5530 . . 3  |-  ( X  e.  A  ->  (
( F  o.  G
) `  X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) ) )
87adantl 277 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( iota
x x  e.  ( ( F  o.  G
) " { X } ) ) )
9 funfvex 5551 . . . 4  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( G `  X
)  e.  _V )
109funfni 5335 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G `  X
)  e.  _V )
11 dffv3g 5530 . . 3  |-  ( ( G `  X )  e.  _V  ->  ( F `  ( G `  X ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
1210, 11syl 14 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F `  ( G `  X )
)  =  ( iota
x x  e.  ( F " { ( G `  X ) } ) ) )
136, 8, 123eqtr4d 2232 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 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   {csn 3607   "cima 4647    o. ccom 4648   iotacio 5194    Fn wfn 5230   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  fvco  5607  fvco3  5608  ofco  6125  updjudhcoinlf  7109  updjudhcoinrg  7110  updjud  7111  caseinl  7120  caseinr  7121  ctm  7138  enomnilem  7166  enmkvlem  7189  enwomnilem  7197  ringidvalg  13315  lidlvalg  13787  rspvalg  13788
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