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Theorem fvco4i 5801
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 5482 . . . 4  |-  ( Fun 
G  <->  G  Fn  dom  G )
31, 2mpbi 200 . . 3  |-  G  Fn  dom  G
4 fvco2 5798 . . 3  |-  ( ( G  Fn  dom  G  /\  X  e.  dom  G )  ->  ( ( F  o.  G ) `  X )  =  ( F `  ( G `
 X ) ) )
53, 4mpan 652 . 2  |-  ( X  e.  dom  G  -> 
( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
6 fvco4i.a . . 3  |-  (/)  =  ( F `  (/) )
7 dmcoss 5135 . . . . . 6  |-  dom  ( F  o.  G )  C_ 
dom  G
87sseli 3344 . . . . 5  |-  ( X  e.  dom  ( F  o.  G )  ->  X  e.  dom  G )
98con3i 129 . . . 4  |-  ( -.  X  e.  dom  G  ->  -.  X  e.  dom  ( F  o.  G
) )
10 ndmfv 5755 . . . 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 5755 . . . 4  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
1312fveq2d 5732 . . 3  |-  ( -.  X  e.  dom  G  ->  ( F `  ( G `  X )
)  =  ( F `
 (/) ) )
146, 11, 133eqtr4a 2494 . 2  |-  ( -.  X  e.  dom  G  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
155, 14pm2.61i 158 1  |-  ( ( F  o.  G ) `
 X )  =  ( F `  ( G `  X )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   (/)c0 3628   dom cdm 4878    o. ccom 4882   Fun wfun 5448    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  lidlval  16265  rspval  16266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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