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Mirrors > Home > ILE Home > Th. List > elreldm | Unicode version |
Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.) |
Ref | Expression |
---|---|
elreldm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 4516 | . . . . 5 | |
2 | ssel 3061 | . . . . 5 | |
3 | 1, 2 | sylbi 120 | . . . 4 |
4 | elvv 4571 | . . . 4 | |
5 | 3, 4 | syl6ib 160 | . . 3 |
6 | eleq1 2180 | . . . . . 6 | |
7 | vex 2663 | . . . . . . 7 | |
8 | vex 2663 | . . . . . . 7 | |
9 | 7, 8 | opeldm 4712 | . . . . . 6 |
10 | 6, 9 | syl6bi 162 | . . . . 5 |
11 | inteq 3744 | . . . . . . . 8 | |
12 | 11 | inteqd 3746 | . . . . . . 7 |
13 | 7, 8 | op1stb 4369 | . . . . . . 7 |
14 | 12, 13 | syl6eq 2166 | . . . . . 6 |
15 | 14 | eleq1d 2186 | . . . . 5 |
16 | 10, 15 | sylibrd 168 | . . . 4 |
17 | 16 | exlimivv 1852 | . . 3 |
18 | 5, 17 | syli 37 | . 2 |
19 | 18 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wex 1453 wcel 1465 cvv 2660 wss 3041 cop 3500 cint 3741 cxp 4507 cdm 4509 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-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-int 3742 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-dm 4519 |
This theorem is referenced by: 1stdm 6048 fundmen 6668 |
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