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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 4880 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | ssel 3141 | . . . . . . . 8 | |
4 | vex 2733 | . . . . . . . . . . 11 | |
5 | 4 | elrn 4854 | . . . . . . . . . 10 |
6 | 4 | eldm 4808 | . . . . . . . . . 10 |
7 | 5, 6 | imbi12i 238 | . . . . . . . . 9 |
8 | 19.8a 1583 | . . . . . . . . . . 11 | |
9 | 8 | imim1i 60 | . . . . . . . . . 10 |
10 | pm3.2 138 | . . . . . . . . . . 11 | |
11 | 10 | eximdv 1873 | . . . . . . . . . 10 |
12 | 9, 11 | sylcom 28 | . . . . . . . . 9 |
13 | 7, 12 | sylbi 120 | . . . . . . . 8 |
14 | 3, 13 | syl 14 | . . . . . . 7 |
15 | 14 | eximdv 1873 | . . . . . 6 |
16 | excom 1657 | . . . . . 6 | |
17 | 15, 16 | syl6ibr 161 | . . . . 5 |
18 | vex 2733 | . . . . . . 7 | |
19 | vex 2733 | . . . . . . 7 | |
20 | 18, 19 | opelco 4783 | . . . . . 6 |
21 | 20 | exbii 1598 | . . . . 5 |
22 | 17, 21 | syl6ibr 161 | . . . 4 |
23 | 18 | eldm 4808 | . . . 4 |
24 | 18 | eldm2 4809 | . . . 4 |
25 | 22, 23, 24 | 3imtr4g 204 | . . 3 |
26 | 25 | ssrdv 3153 | . 2 |
27 | 2, 26 | eqssd 3164 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wex 1485 wcel 2141 wss 3121 cop 3586 class class class wbr 3989 cdm 4611 crn 4612 ccom 4615 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 |
This theorem is referenced by: dmcoeq 4883 fnco 5306 |
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