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Theorem foco 5059
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
foco  F : -onto-> C  G : -onto->  F  o.  G : -onto-> C

Proof of Theorem foco
StepHypRef Expression
1 dffo2 5053 . . 3  F : -onto-> C  F : --> C  ran  F  C
2 dffo2 5053 . . 3  G : -onto->  G : -->  ran  G
3 fco 4999 . . . . 5  F : --> C  G : -->  F  o.  G : --> C
43ad2ant2r 478 . . . 4  F : --> C  ran  F  C  G : -->  ran  G  F  o.  G : --> C
5 fdm 4993 . . . . . . . 8  F : --> C 
dom  F
6 eqtr3 2056 . . . . . . . 8  dom  F  ran  G  dom  F 
ran  G
75, 6sylan 267 . . . . . . 7  F : --> C  ran  G  dom  F  ran  G
8 rncoeq 4548 . . . . . . . . 9  dom 
F  ran  G  ran  F  o.  G  ran  F
98eqeq1d 2045 . . . . . . . 8  dom 
F  ran  G  ran  F  o.  G  C  ran  F  C
109biimpar 281 . . . . . . 7  dom  F  ran  G  ran  F  C  ran  F  o.  G  C
117, 10sylan 267 . . . . . 6  F : --> C  ran  G  ran  F  C 
ran  F  o.  G  C
1211an32s 502 . . . . 5  F : --> C  ran  F  C  ran  G 
ran  F  o.  G  C
1312adantrl 447 . . . 4  F : --> C  ran  F  C  G : -->  ran  G  ran  F  o.  G  C
144, 13jca 290 . . 3  F : --> C  ran  F  C  G : -->  ran  G  F  o.  G : --> C  ran  F  o.  G  C
151, 2, 14syl2anb 275 . 2  F : -onto-> C  G : -onto->  F  o.  G : --> C  ran  F  o.  G  C
16 dffo2 5053 . 2  F  o.  G : -onto-> C  F  o.  G : --> C  ran  F  o.  G  C
1715, 16sylibr 137 1  F : -onto-> C  G : -onto->  F  o.  G : -onto-> C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   dom cdm 4288   ran crn 4289    o. ccom 4292   -->wf 4841   -onto->wfo 4843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851
This theorem is referenced by:  f1oco  5092
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