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Theorem fnco 5368
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 5357 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2 fnfun 5357 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
3 funco 5308 . . . 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 5359 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
76sseq2d 3219 . . . . . 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 4962 . . . . 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 5359 . . . 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 2328 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  B )
15 df-fn 5274 . 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 1632    C_ wss 3165   dom cdm 4705   ran crn 4706    o. ccom 4709   Fun wfun 5265    Fn wfn 5266
This theorem is referenced by:  fco  5414  fnfco  5423  fipreima  7177  prdsinvlem  14619  prdsmgp  15409  pws1  15415  upxp  17333  uptx  17335  evlslem1  19415  0vfval  21178  xppreima2  23227  frlmbas  27326  frlmup3  27355  frlmup4  27356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274
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