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Theorem fvco4i 5958
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 5618 . . . 4  |-  ( Fun 
G  <->  G  Fn  dom  G )
31, 2mpbi 213 . . 3  |-  G  Fn  dom  G
4 fvco2 5955 . . 3  |-  ( ( G  Fn  dom  G  /\  X  e.  dom  G )  ->  ( ( F  o.  G ) `  X )  =  ( F `  ( G `
 X ) ) )
53, 4mpan 684 . 2  |-  ( X  e.  dom  G  -> 
( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
6 fvco4i.a . . 3  |-  (/)  =  ( F `  (/) )
7 dmcoss 5100 . . . . . 6  |-  dom  ( F  o.  G )  C_ 
dom  G
87sseli 3414 . . . . 5  |-  ( X  e.  dom  ( F  o.  G )  ->  X  e.  dom  G )
98con3i 142 . . . 4  |-  ( -.  X  e.  dom  G  ->  -.  X  e.  dom  ( F  o.  G
) )
10 ndmfv 5903 . . . 4  |-  ( -.  X  e.  dom  ( F  o.  G )  ->  ( ( F  o.  G ) `  X
)  =  (/) )
119, 10syl 17 . . 3  |-  ( -.  X  e.  dom  G  ->  ( ( F  o.  G ) `  X
)  =  (/) )
12 ndmfv 5903 . . . 4  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
1312fveq2d 5883 . . 3  |-  ( -.  X  e.  dom  G  ->  ( F `  ( G `  X )
)  =  ( F `
 (/) ) )
146, 11, 133eqtr4a 2531 . 2  |-  ( -.  X  e.  dom  G  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
155, 14pm2.61i 169 1  |-  ( ( F  o.  G ) `
 X )  =  ( F `  ( G `  X )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904   (/)c0 3722   dom cdm 4839    o. ccom 4843   Fun wfun 5583    Fn wfn 5584   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  lidlval  18493  rspval  18494
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