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Theorem fnco 5352
Description: Composition of two functions. (Contributed by NM, 22-May-2006.)
Assertion
Ref Expression
fnco  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )

Proof of Theorem fnco
StepHypRef Expression
1 fnfun 5341 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2 fnfun 5341 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
3 funco 5292 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 463 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  Fun  ( F  o.  G ) )
543adant3 975 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  Fun  ( F  o.  G ) )
6 fndm 5343 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
76sseq2d 3206 . . . . . 6  |-  ( F  Fn  A  ->  ( ran  G  C_  dom  F  <->  ran  G  C_  A ) )
87biimpar 471 . . . . 5  |-  ( ( F  Fn  A  /\  ran  G  C_  A )  ->  ran  G  C_  dom  F )
9 dmcosseq 4946 . . . . 5  |-  ( ran 
G  C_  dom  F  ->  dom  ( F  o.  G
)  =  dom  G
)
108, 9syl 15 . . . 4  |-  ( ( F  Fn  A  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  dom  G )
11103adant2 974 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  dom  G )
12 fndm 5343 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
13123ad2ant2 977 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  G  =  B )
1411, 13eqtrd 2315 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  B )
15 df-fn 5258 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
165, 14, 15sylanbrc 645 1  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  ( F  o.  G
)  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    C_ wss 3152   dom cdm 4689   ran crn 4690    o. ccom 4693   Fun wfun 5249    Fn wfn 5250
This theorem is referenced by:  fco  5398  fnfco  5407  fipreima  7161  prdsinvlem  14603  prdsmgp  15393  pws1  15399  upxp  17317  uptx  17319  evlslem1  19399  0vfval  21162  xppreima2  23212  frlmbas  27223  frlmup3  27252  frlmup4  27253
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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-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-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-fun 5257  df-fn 5258
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