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Theorem dmcosseq 4546
Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmcosseq  ran  C_  dom  dom  o.  dom

Proof of Theorem dmcosseq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 4544 . . 3  dom  o.  C_  dom
21a1i 9 . 2  ran  C_  dom  dom  o.  C_  dom
3 ssel 2933 . . . . . . . 8  ran  C_  dom  ran  dom
4 vex 2554 . . . . . . . . . . 11 
_V
54elrn 4520 . . . . . . . . . 10  ran
64eldm 4475 . . . . . . . . . 10  dom
75, 6imbi12i 228 . . . . . . . . 9  ran  dom
8 19.8a 1479 . . . . . . . . . . 11
98imim1i 54 . . . . . . . . . 10
10 pm3.2 126 . . . . . . . . . . 11
1110eximdv 1757 . . . . . . . . . 10
129, 11sylcom 25 . . . . . . . . 9
137, 12sylbi 114 . . . . . . . 8  ran  dom
143, 13syl 14 . . . . . . 7  ran  C_  dom
1514eximdv 1757 . . . . . 6  ran  C_  dom
16 excom 1551 . . . . . 6
1715, 16syl6ibr 151 . . . . 5  ran  C_  dom
18 vex 2554 . . . . . . 7 
_V
19 vex 2554 . . . . . . 7 
_V
2018, 19opelco 4450 . . . . . 6  <. ,  >.  o.
2120exbii 1493 . . . . 5  <. , 
>.  o.
2217, 21syl6ibr 151 . . . 4  ran  C_  dom  <. ,  >.  o.
2318eldm 4475 . . . 4  dom
2418eldm2 4476 . . . 4  dom  o.  <. ,  >.  o.
2522, 23, 243imtr4g 194 . . 3  ran  C_  dom  dom  dom  o.
2625ssrdv 2945 . 2  ran  C_  dom  dom  C_  dom  o.
272, 26eqssd 2956 1  ran  C_  dom  dom  o.  dom
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390    C_ wss 2911   <.cop 3370   class class class wbr 3755   dom cdm 4288   ran crn 4289    o. ccom 4292
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-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-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299
This theorem is referenced by:  dmcoeq  4547  fnco  4950
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