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Theorem foco 5663
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 5657 . . 3  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  ran  F  =  C ) )
2 dffo2 5657 . . 3  |-  ( G : A -onto-> B  <->  ( G : A --> B  /\  ran  G  =  B ) )
3 fco 5600 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
43ad2ant2r 728 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ( F  o.  G ) : A --> C )
5 fdm 5595 . . . . . . . 8  |-  ( F : B --> C  ->  dom  F  =  B )
6 eqtr3 2455 . . . . . . . 8  |-  ( ( dom  F  =  B  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
75, 6sylan 458 . . . . . . 7  |-  ( ( F : B --> C  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
8 rncoeq 5139 . . . . . . . . 9  |-  ( dom 
F  =  ran  G  ->  ran  ( F  o.  G )  =  ran  F )
98eqeq1d 2444 . . . . . . . 8  |-  ( dom 
F  =  ran  G  ->  ( ran  ( F  o.  G )  =  C  <->  ran  F  =  C ) )
109biimpar 472 . . . . . . 7  |-  ( ( dom  F  =  ran  G  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
117, 10sylan 458 . . . . . 6  |-  ( ( ( F : B --> C  /\  ran  G  =  B )  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
1211an32s 780 . . . . 5  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ran  G  =  B )  ->  ran  ( F  o.  G
)  =  C )
1312adantrl 697 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ran  ( F  o.  G )  =  C )
144, 13jca 519 . . 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 466 . 2  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
16 dffo2 5657 . 2  |-  ( ( F  o.  G ) : A -onto-> C  <->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
1715, 16sylibr 204 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 359    = wceq 1652   dom cdm 4878   ran crn 4879    o. ccom 4882   -->wf 5450   -onto->wfo 5452
This theorem is referenced by:  f1oco  5698  wdomtr  7543  fin1a2lem7  8286  cofull  14131  uniiccdif  19470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460
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