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Theorem fvco4i 5873
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 5550 . . . 4  |-  ( Fun 
G  <->  G  Fn  dom  G )
31, 2mpbi 208 . . 3  |-  G  Fn  dom  G
4 fvco2 5870 . . 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 5202 . . . . . 6  |-  dom  ( F  o.  G )  C_ 
dom  G
87sseli 3455 . . . . 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 5818 . . . 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 5818 . . . 4  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
1312fveq2d 5798 . . 3  |-  ( -.  X  e.  dom  G  ->  ( F `  ( G `  X )
)  =  ( F `
 (/) ) )
146, 11, 133eqtr4a 2519 . 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 1370    e. wcel 1758   (/)c0 3740   dom cdm 4943    o. ccom 4947   Fun wfun 5515    Fn wfn 5516   ` cfv 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-fv 5529
This theorem is referenced by:  lidlval  17391  rspval  17392
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