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Theorem fvco2 5270
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 fnsnfv 5260 . . . . . 6  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
21imaeq2d 4696 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( F " {
( G `  X
) } )  =  ( F " ( G " { X }
) ) )
3 imaco 4854 . . . . 5  |-  ( ( F  o.  G )
" { X }
)  =  ( F
" ( G " { X } ) )
42, 3syl6reqr 2107 . . . 4  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) " { X } )  =  ( F " { ( G `  X ) } ) )
54eleq2d 2123 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( x  e.  ( ( F  o.  G
) " { X } )  <->  x  e.  ( F " { ( G `  X ) } ) ) )
65iotabidv 4916 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( iota x x  e.  ( ( F  o.  G ) " { X } ) )  =  ( iota x x  e.  ( F " { ( G `  X ) } ) ) )
7 dffv3g 5202 . . 3  |-  ( X  e.  A  ->  (
( F  o.  G
) `  X )  =  ( iota x x  e.  ( ( F  o.  G ) " { X } ) ) )
87adantl 266 . 2  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( ( F  o.  G ) `  X
)  =  ( iota
x x  e.  ( ( F  o.  G
) " { X } ) ) )
9 funfvex 5220 . . . 4  |-  ( ( Fun  G  /\  X  e.  dom  G )  -> 
( G `  X
)  e.  _V )
109funfni 5027 . . 3  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G `  X
)  e.  _V )
11 dffv3g 5202 . . 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 2098 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 101    = wceq 1259    e. wcel 1409   _Vcvv 2574   {csn 3403   "cima 4376    o. ccom 4377   iotacio 4893    Fn wfn 4925   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by:  fvco  5271  fvco3  5272  ofco  5757
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