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Theorem fnco 5201
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 5190 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2 fnfun 5190 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
3 funco 5133 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 287 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  Fun  ( F  o.  G ) )
543adant3 986 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  Fun  ( F  o.  G ) )
6 fndm 5192 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
76sseq2d 3097 . . . . . 6  |-  ( F  Fn  A  ->  ( ran  G  C_  dom  F  <->  ran  G  C_  A ) )
87biimpar 295 . . . . 5  |-  ( ( F  Fn  A  /\  ran  G  C_  A )  ->  ran  G  C_  dom  F )
9 dmcosseq 4780 . . . . 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 985 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  dom  G )
12 fndm 5192 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
13123ad2ant2 988 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  G  =  B )
1411, 13eqtrd 2150 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  B )
15 df-fn 5096 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
165, 14, 15sylanbrc 413 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 103    /\ w3a 947    = wceq 1316    C_ wss 3041   dom cdm 4509   ran crn 4510    o. ccom 4513   Fun wfun 5087    Fn wfn 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096
This theorem is referenced by:  fco  5258  fnfco  5267  updjudhcoinlf  6933  updjudhcoinrg  6934  upxp  12352  uptx  12354
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