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Mirrors > Home > ILE Home > Th. List > dmtpos | Unicode version |
Description: The domain of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
dmtpos | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 4626 | . . . . 5 | |
2 | ssel 3131 | . . . . 5 | |
3 | 1, 2 | mtoi 654 | . . . 4 |
4 | df-rel 4605 | . . . 4 | |
5 | reldmtpos 6212 | . . . 4 tpos | |
6 | 3, 4, 5 | 3imtr4i 200 | . . 3 tpos |
7 | relcnv 4976 | . . 3 | |
8 | 6, 7 | jctir 311 | . 2 tpos |
9 | vex 2724 | . . . . . . 7 | |
10 | vex 2724 | . . . . . . 7 | |
11 | vex 2724 | . . . . . . 7 | |
12 | brtposg 6213 | . . . . . . 7 tpos | |
13 | 9, 10, 11, 12 | mp3an 1326 | . . . . . 6 tpos |
14 | 13 | a1i 9 | . . . . 5 tpos |
15 | 14 | exbidv 1812 | . . . 4 tpos |
16 | 9, 10 | opex 4201 | . . . . 5 |
17 | 16 | eldm 4795 | . . . 4 tpos tpos |
18 | 9, 10 | opelcnv 4780 | . . . . 5 |
19 | 10, 9 | opex 4201 | . . . . . 6 |
20 | 19 | eldm 4795 | . . . . 5 |
21 | 18, 20 | bitri 183 | . . . 4 |
22 | 15, 17, 21 | 3bitr4g 222 | . . 3 tpos |
23 | 22 | eqrelrdv2 4697 | . 2 tpos tpos |
24 | 8, 23 | mpancom 419 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2721 wss 3111 c0 3404 cop 3573 class class class wbr 3976 cxp 4596 ccnv 4597 cdm 4598 wrel 4603 tpos ctpos 6203 |
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-in1 604 ax-in2 605 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-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-tpos 6204 |
This theorem is referenced by: rntpos 6216 dftpos2 6220 dftpos3 6221 tposfn2 6225 |
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