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Mirrors > Home > ILE Home > Th. List > dmsnopg | Unicode version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2684 | . . . . . 6 | |
2 | vex 2684 | . . . . . 6 | |
3 | 1, 2 | opth1 4153 | . . . . 5 |
4 | 3 | exlimiv 1577 | . . . 4 |
5 | opeq1 3700 | . . . . 5 | |
6 | opeq2 3701 | . . . . . . 7 | |
7 | 6 | eqeq1d 2146 | . . . . . 6 |
8 | 7 | spcegv 2769 | . . . . 5 |
9 | 5, 8 | syl5 32 | . . . 4 |
10 | 4, 9 | impbid2 142 | . . 3 |
11 | 1 | eldm2 4732 | . . . 4 |
12 | 1, 2 | opex 4146 | . . . . . 6 |
13 | 12 | elsn 3538 | . . . . 5 |
14 | 13 | exbii 1584 | . . . 4 |
15 | 11, 14 | bitri 183 | . . 3 |
16 | velsn 3539 | . . 3 | |
17 | 10, 15, 16 | 3bitr4g 222 | . 2 |
18 | 17 | eqrdv 2135 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wex 1468 wcel 1480 csn 3522 cop 3525 cdm 4534 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-dm 4544 |
This theorem is referenced by: dmpropg 5006 dmsnop 5007 rnsnopg 5012 elxp4 5021 fnsng 5165 funprg 5168 funtpg 5169 fntpg 5174 ennnfonelemhdmp1 11911 ennnfonelemkh 11914 setsvala 11979 setsresg 11986 setscom 11988 setsslid 11998 strle1g 12038 |
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