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Mirrors > Home > ILE Home > Th. List > dmrnssfld | Unicode version |
Description: The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.) |
Ref | Expression |
---|---|
dmrnssfld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . . . 5 | |
2 | 1 | eldm2 4737 | . . . 4 |
3 | 1 | prid1 3629 | . . . . . 6 |
4 | vex 2689 | . . . . . . . . . 10 | |
5 | 1, 4 | uniop 4177 | . . . . . . . . 9 |
6 | 1, 4 | uniopel 4178 | . . . . . . . . 9 |
7 | 5, 6 | eqeltrrid 2227 | . . . . . . . 8 |
8 | elssuni 3764 | . . . . . . . 8 | |
9 | 7, 8 | syl 14 | . . . . . . 7 |
10 | 9 | sseld 3096 | . . . . . 6 |
11 | 3, 10 | mpi 15 | . . . . 5 |
12 | 11 | exlimiv 1577 | . . . 4 |
13 | 2, 12 | sylbi 120 | . . 3 |
14 | 13 | ssriv 3101 | . 2 |
15 | 4 | elrn2 4781 | . . . 4 |
16 | 4 | prid2 3630 | . . . . . 6 |
17 | 9 | sseld 3096 | . . . . . 6 |
18 | 16, 17 | mpi 15 | . . . . 5 |
19 | 18 | exlimiv 1577 | . . . 4 |
20 | 15, 19 | sylbi 120 | . . 3 |
21 | 20 | ssriv 3101 | . 2 |
22 | 14, 21 | unssi 3251 | 1 |
Colors of variables: wff set class |
Syntax hints: wex 1468 wcel 1480 cun 3069 wss 3071 cpr 3528 cop 3530 cuni 3736 cdm 4539 crn 4540 |
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-rex 2422 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-uni 3737 df-br 3930 df-opab 3990 df-cnv 4547 df-dm 4549 df-rn 4550 |
This theorem is referenced by: dmexg 4803 rnexg 4804 relfld 5067 relcoi2 5069 |
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