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Mirrors > Home > ILE Home > Th. List > opeldmg | Unicode version |
Description: Membership of first of an ordered pair in a domain. (Contributed by Jim Kingdon, 9-Jul-2019.) |
Ref | Expression |
---|---|
opeldmg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 3571 | . . . . 5 | |
2 | 1 | eleq1d 2147 | . . . 4 |
3 | 2 | spcegv 2686 | . . 3 |
4 | 3 | adantl 271 | . 2 |
5 | eldm2g 4549 | . . 3 | |
6 | 5 | adantr 270 | . 2 |
7 | 4, 6 | sylibrd 167 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 cop 3401 cdm 4363 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-dm 4373 |
This theorem is referenced by: tfr0 5960 tfrlemi14d 5970 fnfi 6388 frecuzrdgfn 9414 |
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