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Theorem foco 5817
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 5811 . . 3  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  ran  F  =  C ) )
2 dffo2 5811 . . 3  |-  ( G : A -onto-> B  <->  ( G : A --> B  /\  ran  G  =  B ) )
3 fco 5753 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
43ad2ant2r 751 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ( F  o.  G ) : A --> C )
5 fdm 5747 . . . . . . . 8  |-  ( F : B --> C  ->  dom  F  =  B )
6 eqtr3 2450 . . . . . . . 8  |-  ( ( dom  F  =  B  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
75, 6sylan 473 . . . . . . 7  |-  ( ( F : B --> C  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
8 rncoeq 5114 . . . . . . . . 9  |-  ( dom 
F  =  ran  G  ->  ran  ( F  o.  G )  =  ran  F )
98eqeq1d 2424 . . . . . . . 8  |-  ( dom 
F  =  ran  G  ->  ( ran  ( F  o.  G )  =  C  <->  ran  F  =  C ) )
109biimpar 487 . . . . . . 7  |-  ( ( dom  F  =  ran  G  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
117, 10sylan 473 . . . . . 6  |-  ( ( ( F : B --> C  /\  ran  G  =  B )  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
1211an32s 811 . . . . 5  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ran  G  =  B )  ->  ran  ( F  o.  G
)  =  C )
1312adantrl 720 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ran  ( F  o.  G )  =  C )
144, 13jca 534 . . 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 481 . 2  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
16 dffo2 5811 . 2  |-  ( ( F  o.  G ) : A -onto-> C  <->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
1715, 16sylibr 215 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 370    = wceq 1437   dom cdm 4850   ran crn 4851    o. ccom 4854   -->wf 5594   -onto->wfo 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-fun 5600  df-fn 5601  df-f 5602  df-fo 5604
This theorem is referenced by:  f1oco  5850  wdomtr  8093  fin1a2lem7  8837  cofull  15827  uniiccdif  22522
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