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Mirrors > Home > ILE Home > Th. List > relopabi | Unicode version |
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
relopabi.1 |
Ref | Expression |
---|---|
relopabi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopabi.1 | . . . 4 | |
2 | df-opab 3960 | . . . 4 | |
3 | 1, 2 | eqtri 2138 | . . 3 |
4 | vex 2663 | . . . . . . . 8 | |
5 | vex 2663 | . . . . . . . 8 | |
6 | 4, 5 | opelvv 4559 | . . . . . . 7 |
7 | eleq1 2180 | . . . . . . 7 | |
8 | 6, 7 | mpbiri 167 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | 9 | exlimivv 1852 | . . . 4 |
11 | 10 | abssi 3142 | . . 3 |
12 | 3, 11 | eqsstri 3099 | . 2 |
13 | df-rel 4516 | . 2 | |
14 | 12, 13 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1316 wex 1453 wcel 1465 cab 2103 cvv 2660 wss 3041 cop 3500 copab 3958 cxp 4507 wrel 4514 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-xp 4515 df-rel 4516 |
This theorem is referenced by: relopab 4636 mptrel 4637 reli 4638 rele 4639 relcnv 4887 cotr 4890 relco 5007 reloprab 5787 reldmoprab 5824 eqer 6429 ecopover 6495 ecopoverg 6498 relen 6606 reldom 6607 enq0er 7211 aprcl 8375 climrel 11004 brstruct 11879 |
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