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Theorem fnco 5199
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 5188 . . . 4  |-  ( F  Fn  A  ->  Fun  F )
2 fnfun 5188 . . . 4  |-  ( G  Fn  B  ->  Fun  G )
3 funco 5131 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 285 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  Fun  ( F  o.  G ) )
543adant3 984 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  Fun  ( F  o.  G ) )
6 fndm 5190 . . . . . . 7  |-  ( F  Fn  A  ->  dom  F  =  A )
76sseq2d 3095 . . . . . 6  |-  ( F  Fn  A  ->  ( ran  G  C_  dom  F  <->  ran  G  C_  A ) )
87biimpar 293 . . . . 5  |-  ( ( F  Fn  A  /\  ran  G  C_  A )  ->  ran  G  C_  dom  F )
9 dmcosseq 4778 . . . . 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 983 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  dom  G )
12 fndm 5190 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
13123ad2ant2 986 . . 3  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  G  =  B )
1411, 13eqtrd 2148 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B  /\  ran  G  C_  A )  ->  dom  ( F  o.  G )  =  B )
15 df-fn 5094 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
165, 14, 15sylanbrc 411 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 945    = wceq 1314    C_ wss 3039   dom cdm 4507   ran crn 4508    o. ccom 4511   Fun wfun 5085    Fn wfn 5086
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-fun 5093  df-fn 5094
This theorem is referenced by:  fco  5256  fnfco  5265  updjudhcoinlf  6931  updjudhcoinrg  6932  upxp  12347  uptx  12349
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