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Theorem fnco 5108
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 5097 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2 fnfun 5097 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
3 funco 5040 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 283 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  Fun  ( F  o.  G ) )
543adant3 963 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  Fun  ( F  o.  G ) )
6 fndm 5099 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
76sseq2d 3052 . . . . . 6  |-  ( F  Fn  A  ->  ( ran  G  C_  dom  F  <->  ran  G  C_  A ) )
87biimpar 291 . . . . 5  |-  ( ( F  Fn  A  /\  ran  G  C_  A )  ->  ran  G  C_  dom  F )
9 dmcosseq 4692 . . . . 5  |-  ( ran 
G  C_  dom  F  ->  dom  ( F  o.  G
)  =  dom  G
)
108, 9syl 14 . . . 4  |-  ( ( F  Fn  A  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  dom  G )
11103adant2 962 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  dom  G )
12 fndm 5099 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
13123ad2ant2 965 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  G  =  B )
1411, 13eqtrd 2120 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  B )
15 df-fn 5005 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
165, 14, 15sylanbrc 408 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 102    /\ w3a 924    = wceq 1289    C_ wss 2997   dom cdm 4428   ran crn 4429    o. ccom 4432   Fun wfun 4996    Fn wfn 4997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005
This theorem is referenced by:  fco  5161  fnfco  5170  updjudhcoinlf  6750  updjudhcoinrg  6751
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