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Mirrors > Home > ILE Home > Th. List > releldm2 | Unicode version |
Description: Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
Ref | Expression |
---|---|
releldm2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2671 | . . 3 | |
2 | 1 | anim2i 339 | . 2 |
3 | id 19 | . . . . 5 | |
4 | vex 2663 | . . . . . 6 | |
5 | 1stexg 6033 | . . . . . 6 | |
6 | 4, 5 | ax-mp 5 | . . . . 5 |
7 | 3, 6 | syl6eqelr 2209 | . . . 4 |
8 | 7 | rexlimivw 2522 | . . 3 |
9 | 8 | anim2i 339 | . 2 |
10 | eldm2g 4705 | . . . 4 | |
11 | 10 | adantl 275 | . . 3 |
12 | df-rel 4516 | . . . . . . . . 9 | |
13 | ssel 3061 | . . . . . . . . 9 | |
14 | 12, 13 | sylbi 120 | . . . . . . . 8 |
15 | 14 | imp 123 | . . . . . . 7 |
16 | op1steq 6045 | . . . . . . 7 | |
17 | 15, 16 | syl 14 | . . . . . 6 |
18 | 17 | rexbidva 2411 | . . . . 5 |
19 | 18 | adantr 274 | . . . 4 |
20 | rexcom4 2683 | . . . . 5 | |
21 | risset 2440 | . . . . . 6 | |
22 | 21 | exbii 1569 | . . . . 5 |
23 | 20, 22 | bitr4i 186 | . . . 4 |
24 | 19, 23 | syl6bb 195 | . . 3 |
25 | 11, 24 | bitr4d 190 | . 2 |
26 | 2, 9, 25 | pm5.21nd 886 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 wrex 2394 cvv 2660 wss 3041 cop 3500 cxp 4507 cdm 4509 wrel 4514 cfv 5093 c1st 6004 |
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-13 1476 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 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-1st 6006 df-2nd 6007 |
This theorem is referenced by: reldm 6052 |
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