Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4808 | . . 3 | |
2 | 1 | a1i 9 | . 2 |
3 | ssel 3091 | . . . . . . . 8 | |
4 | vex 2689 | . . . . . . . . . . 11 | |
5 | 4 | elrn 4782 | . . . . . . . . . 10 |
6 | 4 | eldm 4736 | . . . . . . . . . 10 |
7 | 5, 6 | imbi12i 238 | . . . . . . . . 9 |
8 | 19.8a 1569 | . . . . . . . . . . 11 | |
9 | 8 | imim1i 60 | . . . . . . . . . 10 |
10 | pm3.2 138 | . . . . . . . . . . 11 | |
11 | 10 | eximdv 1852 | . . . . . . . . . 10 |
12 | 9, 11 | sylcom 28 | . . . . . . . . 9 |
13 | 7, 12 | sylbi 120 | . . . . . . . 8 |
14 | 3, 13 | syl 14 | . . . . . . 7 |
15 | 14 | eximdv 1852 | . . . . . 6 |
16 | excom 1642 | . . . . . 6 | |
17 | 15, 16 | syl6ibr 161 | . . . . 5 |
18 | vex 2689 | . . . . . . 7 | |
19 | vex 2689 | . . . . . . 7 | |
20 | 18, 19 | opelco 4711 | . . . . . 6 |
21 | 20 | exbii 1584 | . . . . 5 |
22 | 17, 21 | syl6ibr 161 | . . . 4 |
23 | 18 | eldm 4736 | . . . 4 |
24 | 18 | eldm2 4737 | . . . 4 |
25 | 22, 23, 24 | 3imtr4g 204 | . . 3 |
26 | 25 | ssrdv 3103 | . 2 |
27 | 2, 26 | eqssd 3114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wex 1468 wcel 1480 wss 3071 cop 3530 class class class wbr 3929 cdm 4539 crn 4540 ccom 4543 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 |
This theorem is referenced by: dmcoeq 4811 fnco 5231 |
Copyright terms: Public domain | W3C validator |