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Theorem foco 5227
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
foco  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( F  o.  G ) : A -onto-> C )

Proof of Theorem foco
StepHypRef Expression
1 dffo2 5221 . . 3  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  ran  F  =  C ) )
2 dffo2 5221 . . 3  |-  ( G : A -onto-> B  <->  ( G : A --> B  /\  ran  G  =  B ) )
3 fco 5161 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
43ad2ant2r 493 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ( F  o.  G ) : A --> C )
5 fdm 5152 . . . . . . . 8  |-  ( F : B --> C  ->  dom  F  =  B )
6 eqtr3 2107 . . . . . . . 8  |-  ( ( dom  F  =  B  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
75, 6sylan 277 . . . . . . 7  |-  ( ( F : B --> C  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
8 rncoeq 4694 . . . . . . . . 9  |-  ( dom 
F  =  ran  G  ->  ran  ( F  o.  G )  =  ran  F )
98eqeq1d 2096 . . . . . . . 8  |-  ( dom 
F  =  ran  G  ->  ( ran  ( F  o.  G )  =  C  <->  ran  F  =  C ) )
109biimpar 291 . . . . . . 7  |-  ( ( dom  F  =  ran  G  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
117, 10sylan 277 . . . . . 6  |-  ( ( ( F : B --> C  /\  ran  G  =  B )  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
1211an32s 535 . . . . 5  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ran  G  =  B )  ->  ran  ( F  o.  G
)  =  C )
1312adantrl 462 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ran  ( F  o.  G )  =  C )
144, 13jca 300 . . 3  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
151, 2, 14syl2anb 285 . 2  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
16 dffo2 5221 . 2  |-  ( ( F  o.  G ) : A -onto-> C  <->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
1715, 16sylibr 132 1  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( F  o.  G ) : A -onto-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   dom cdm 4428   ran crn 4429    o. ccom 4432   -->wf 4998   -onto->wfo 5000
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  df-f 5006  df-fo 5008
This theorem is referenced by:  f1oco  5260
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