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Theorem fvco2 5555
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 5109 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
2 fnsnfv 5545 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
32imaeq2d 4946 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
41, 3eqtr4id 2218 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2236 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 5174 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3g 5482 . . 3  |-  ( X  e.  A  ->  (
( F  o.  G
) `  X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) ) )
87adantl 275 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( iota
x x  e.  ( ( F  o.  G
) " { X } ) ) )
9 funfvex 5503 . . . 4  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( G `  X
)  e.  _V )
109funfni 5288 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G `  X
)  e.  _V )
11 dffv3g 5482 . . 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 2208 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 103    = wceq 1343    e. wcel 2136   _Vcvv 2726   {csn 3576   "cima 4607    o. ccom 4608   iotacio 5151    Fn wfn 5183   ` cfv 5188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196
This theorem is referenced by:  fvco  5556  fvco3  5557  ofco  6068  updjudhcoinlf  7045  updjudhcoinrg  7046  updjud  7047  caseinl  7056  caseinr  7057  ctm  7074  enomnilem  7102  enmkvlem  7125  enwomnilem  7133
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