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Mirrors > Home > ILE Home > Th. List > dmcosseq | Unicode version |
Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmcosseq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmcoss 4855 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | ssel 3122 | . . . . . . . 8 | |
4 | vex 2715 | . . . . . . . . . . 11 | |
5 | 4 | elrn 4829 | . . . . . . . . . 10 |
6 | 4 | eldm 4783 | . . . . . . . . . 10 |
7 | 5, 6 | imbi12i 238 | . . . . . . . . 9 |
8 | 19.8a 1570 | . . . . . . . . . . 11 | |
9 | 8 | imim1i 60 | . . . . . . . . . 10 |
10 | pm3.2 138 | . . . . . . . . . . 11 | |
11 | 10 | eximdv 1860 | . . . . . . . . . 10 |
12 | 9, 11 | sylcom 28 | . . . . . . . . 9 |
13 | 7, 12 | sylbi 120 | . . . . . . . 8 |
14 | 3, 13 | syl 14 | . . . . . . 7 |
15 | 14 | eximdv 1860 | . . . . . 6 |
16 | excom 1644 | . . . . . 6 | |
17 | 15, 16 | syl6ibr 161 | . . . . 5 |
18 | vex 2715 | . . . . . . 7 | |
19 | vex 2715 | . . . . . . 7 | |
20 | 18, 19 | opelco 4758 | . . . . . 6 |
21 | 20 | exbii 1585 | . . . . 5 |
22 | 17, 21 | syl6ibr 161 | . . . 4 |
23 | 18 | eldm 4783 | . . . 4 |
24 | 18 | eldm2 4784 | . . . 4 |
25 | 22, 23, 24 | 3imtr4g 204 | . . 3 |
26 | 25 | ssrdv 3134 | . 2 |
27 | 2, 26 | eqssd 3145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wex 1472 wcel 2128 wss 3102 cop 3563 class class class wbr 3965 cdm 4586 crn 4587 ccom 4590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3966 df-opab 4026 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 |
This theorem is referenced by: dmcoeq 4858 fnco 5278 |
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