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Theorem fco 4999
Description: Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fco  F : --> C  G : -->  F  o.  G : --> C

Proof of Theorem fco
StepHypRef Expression
1 df-f 4849 . . 3  F : --> C  F  Fn  ran  F 
C_  C
2 df-f 4849 . . 3  G : -->  G  Fn  ran  G 
C_
3 fnco 4950 . . . . . . 7  F  Fn  G  Fn  ran  G  C_  F  o.  G  Fn
433expib 1106 . . . . . 6  F  Fn  G  Fn  ran  G  C_  F  o.  G  Fn
54adantr 261 . . . . 5  F  Fn  ran  F  C_  C  G  Fn  ran  G  C_  F  o.  G  Fn
6 rncoss 4545 . . . . . . 7  ran  F  o.  G  C_ 
ran  F
7 sstr 2947 . . . . . . 7  ran  F  o.  G  C_  ran  F  ran  F  C_  C  ran  F  o.  G  C_  C
86, 7mpan 400 . . . . . 6  ran 
F  C_  C  ran  F  o.  G  C_  C
98adantl 262 . . . . 5  F  Fn  ran  F  C_  C  ran  F  o.  G  C_  C
105, 9jctird 300 . . . 4  F  Fn  ran  F  C_  C  G  Fn  ran  G  C_  F  o.  G  Fn  ran  F  o.  G  C_  C
1110imp 115 . . 3  F  Fn  ran  F  C_  C  G  Fn  ran  G  C_  F  o.  G  Fn  ran  F  o.  G  C_  C
121, 2, 11syl2anb 275 . 2  F : --> C  G : -->  F  o.  G  Fn  ran  F  o.  G  C_  C
13 df-f 4849 . 2  F  o.  G : --> C  F  o.  G  Fn  ran  F  o.  G  C_  C
1412, 13sylibr 137 1  F : --> C  G : -->  F  o.  G : --> C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97    C_ wss 2911   ran crn 4289    o. ccom 4292    Fn wfn 4840   -->wf 4841
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
This theorem is referenced by:  fco2  5000  f1co  5044  foco  5059
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