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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 3706 | . . . . 5 | |
2 | 1 | eleq1d 2208 | . . . 4 |
3 | 2 | spcegv 2774 | . . 3 |
4 | 3 | adantl 275 | . 2 |
5 | eldm2g 4735 | . . 3 | |
6 | 5 | adantr 274 | . 2 |
7 | 4, 6 | sylibrd 168 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3530 cdm 4539 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-dm 4549 |
This theorem is referenced by: tfr0dm 6219 tfrlemi14d 6230 tfr1onlemres 6246 tfrcllemres 6259 fnfi 6825 frecuzrdgtcl 10185 frecuzrdgdomlem 10190 hashennn 10526 |
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