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Theorem fvco4i 5597
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 5283 . . . 4  |-  ( Fun 
G  <->  G  Fn  dom  G )
31, 2mpbi 199 . . 3  |-  G  Fn  dom  G
4 fvco2 5594 . . 3  |-  ( ( G  Fn  dom  G  /\  X  e.  dom  G )  ->  ( ( F  o.  G ) `  X )  =  ( F `  ( G `
 X ) ) )
53, 4mpan 651 . 2  |-  ( X  e.  dom  G  -> 
( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
6 fvco4i.a . . 3  |-  (/)  =  ( F `  (/) )
7 dmcoss 4944 . . . . . 6  |-  dom  ( F  o.  G )  C_ 
dom  G
87sseli 3176 . . . . 5  |-  ( X  e.  dom  ( F  o.  G )  ->  X  e.  dom  G )
98con3i 127 . . . 4  |-  ( -.  X  e.  dom  G  ->  -.  X  e.  dom  ( F  o.  G
) )
10 ndmfv 5552 . . . 4  |-  ( -.  X  e.  dom  ( F  o.  G )  ->  ( ( F  o.  G ) `  X
)  =  (/) )
119, 10syl 15 . . 3  |-  ( -.  X  e.  dom  G  ->  ( ( F  o.  G ) `  X
)  =  (/) )
12 ndmfv 5552 . . . 4  |-  ( -.  X  e.  dom  G  ->  ( G `  X
)  =  (/) )
1312fveq2d 5529 . . 3  |-  ( -.  X  e.  dom  G  ->  ( F `  ( G `  X )
)  =  ( F `
 (/) ) )
146, 11, 133eqtr4a 2341 . 2  |-  ( -.  X  e.  dom  G  ->  ( ( F  o.  G ) `  X
)  =  ( F `
 ( G `  X ) ) )
155, 14pm2.61i 156 1  |-  ( ( F  o.  G ) `
 X )  =  ( F `  ( G `  X )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   (/)c0 3455   dom cdm 4689    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  lidlval  15946  rspval  15947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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