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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 4605 | . . . . 5 | |
2 | ssel 3131 | . . . . 5 | |
3 | 1, 2 | sylbi 120 | . . . 4 |
4 | elvv 4660 | . . . 4 | |
5 | 3, 4 | syl6ib 160 | . . 3 |
6 | eleq1 2227 | . . . . . 6 | |
7 | vex 2724 | . . . . . . 7 | |
8 | vex 2724 | . . . . . . 7 | |
9 | 7, 8 | opeldm 4801 | . . . . . 6 |
10 | 6, 9 | syl6bi 162 | . . . . 5 |
11 | inteq 3821 | . . . . . . . 8 | |
12 | 11 | inteqd 3823 | . . . . . . 7 |
13 | 7, 8 | op1stb 4450 | . . . . . . 7 |
14 | 12, 13 | eqtrdi 2213 | . . . . . 6 |
15 | 14 | eleq1d 2233 | . . . . 5 |
16 | 10, 15 | sylibrd 168 | . . . 4 |
17 | 16 | exlimivv 1883 | . . 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 1342 wex 1479 wcel 2135 cvv 2721 wss 3111 cop 3573 cint 3818 cxp 4596 cdm 4598 wrel 4603 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-int 3819 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-dm 4608 |
This theorem is referenced by: 1stdm 6142 fundmen 6763 |
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