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Theorem fvco4i 5951
Description: Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
Hypotheses
Ref Expression
fvco4i.a  |-  (/)  =  ( F `  (/) )
fvco4i.b  |-  Fun  G
Assertion
Ref Expression
fvco4i  |-  ( ( F  o.  G ) `
 X )  =  ( F `  ( G `  X )
)

Proof of Theorem fvco4i
StepHypRef Expression
1 fvco4i.b . . . 4  |-  Fun  G
2 funfn 5623 . . . 4  |-  ( Fun 
G  <->  G  Fn  dom  G )
31, 2mpbi 208 . . 3  |-  G  Fn  dom  G
4 fvco2 5948 . . 3  |-  ( ( G  Fn  dom  G  /\  X  e.  dom  G )  ->  ( ( F  o.  G ) `  X )  =  ( F `  ( G `
 X ) ) )
53, 4mpan 670 . 2  |-  ( X  e.  dom  G  -> 
( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
6 fvco4i.a . . 3  |-  (/)  =  ( F `  (/) )
7 dmcoss 5272 . . . . . 6  |-  dom  ( F  o.  G )  C_ 
dom  G
87sseli 3495 . . . . 5  |-  ( X  e.  dom  ( F  o.  G )  ->  X  e.  dom  G )
98con3i 135 . . . 4  |-  ( -.  X  e.  dom  G  ->  -.  X  e.  dom  ( F  o.  G
) )
10 ndmfv 5896 . . . 4  |-  ( -.  X  e.  dom  ( F  o.  G )  ->  ( ( F  o.  G ) `  X
)  =  (/) )
119, 10syl 16 . . 3  |-  ( -.  X  e.  dom  G  ->  ( ( F  o.  G ) `  X
)  =  (/) )
12 ndmfv 5896 . . . 4  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
1312fveq2d 5876 . . 3  |-  ( -.  X  e.  dom  G  ->  ( F `  ( G `  X )
)  =  ( F `
 (/) ) )
146, 11, 133eqtr4a 2524 . 2  |-  ( -.  X  e.  dom  G  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
155, 14pm2.61i 164 1  |-  ( ( F  o.  G ) `
 X )  =  ( F `  ( G `  X )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   (/)c0 3793   dom cdm 5008    o. ccom 5012   Fun wfun 5588    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  lidlval  17965  rspval  17966
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