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Theorem dmcosseq 5138
 Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmcosseq

Proof of Theorem dmcosseq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 5136 . . 3
21a1i 11 . 2
3 ssel 3343 . . . . . . . 8
4 vex 2960 . . . . . . . . . . 11
54elrn 5111 . . . . . . . . . 10
64eldm 5068 . . . . . . . . . 10
75, 6imbi12i 318 . . . . . . . . 9
8 19.8a 1763 . . . . . . . . . . 11
98imim1i 57 . . . . . . . . . 10
10 pm3.2 436 . . . . . . . . . . 11
1110eximdv 1633 . . . . . . . . . 10
129, 11sylcom 28 . . . . . . . . 9
137, 12sylbi 189 . . . . . . . 8
143, 13syl 16 . . . . . . 7
1514eximdv 1633 . . . . . 6
16 excom 1757 . . . . . 6
1715, 16syl6ibr 220 . . . . 5
18 vex 2960 . . . . . . 7
19 vex 2960 . . . . . . 7
2018, 19opelco 5045 . . . . . 6
2120exbii 1593 . . . . 5
2217, 21syl6ibr 220 . . . 4
2318eldm 5068 . . . 4
2418eldm2 5069 . . . 4
2522, 23, 243imtr4g 263 . . 3
2625ssrdv 3355 . 2
272, 26eqssd 3366 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wex 1551   wceq 1653   wcel 1726   wss 3321  cop 3818   class class class wbr 4213   cdm 4879   crn 4880   ccom 4883 This theorem is referenced by:  dmcoeq  5139  fnco  5554  fnresfnco  27967 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890
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