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Theorem foco 5816
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 5810 . . 3  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  ran  F  =  C ) )
2 dffo2 5810 . . 3  |-  ( G : A -onto-> B  <->  ( G : A --> B  /\  ran  G  =  B ) )
3 fco 5751 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
43ad2ant2r 761 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ( F  o.  G ) : A --> C )
5 fdm 5745 . . . . . . . 8  |-  ( F : B --> C  ->  dom  F  =  B )
6 eqtr3 2492 . . . . . . . 8  |-  ( ( dom  F  =  B  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
75, 6sylan 479 . . . . . . 7  |-  ( ( F : B --> C  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
8 rncoeq 5104 . . . . . . . . 9  |-  ( dom 
F  =  ran  G  ->  ran  ( F  o.  G )  =  ran  F )
98eqeq1d 2473 . . . . . . . 8  |-  ( dom 
F  =  ran  G  ->  ( ran  ( F  o.  G )  =  C  <->  ran  F  =  C ) )
109biimpar 493 . . . . . . 7  |-  ( ( dom  F  =  ran  G  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
117, 10sylan 479 . . . . . 6  |-  ( ( ( F : B --> C  /\  ran  G  =  B )  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
1211an32s 821 . . . . 5  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ran  G  =  B )  ->  ran  ( F  o.  G
)  =  C )
1312adantrl 730 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ran  ( F  o.  G )  =  C )
144, 13jca 541 . . 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 487 . 2  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
16 dffo2 5810 . 2  |-  ( ( F  o.  G ) : A -onto-> C  <->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
1715, 16sylibr 217 1  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( F  o.  G ) : A -onto-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452   dom cdm 4839   ran crn 4840    o. ccom 4843   -->wf 5585   -onto->wfo 5587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595
This theorem is referenced by:  f1oco  5850  wdomtr  8108  fin1a2lem7  8854  cofull  15917  uniiccdif  22614
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