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Mirrors > Home > ILE Home > Th. List > dftpos3 | Unicode version |
Description: Alternate definition of tpos when has relational domain. Compare df-cnv 4547. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
dftpos3 | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4917 | . . . . . . . . . 10 | |
2 | dmtpos 6153 | . . . . . . . . . . 11 tpos | |
3 | 2 | releqd 4623 | . . . . . . . . . 10 tpos |
4 | 1, 3 | mpbiri 167 | . . . . . . . . 9 tpos |
5 | reltpos 6147 | . . . . . . . . 9 tpos | |
6 | 4, 5 | jctil 310 | . . . . . . . 8 tpos tpos |
7 | relrelss 5065 | . . . . . . . 8 tpos tpos tpos | |
8 | 6, 7 | sylib 121 | . . . . . . 7 tpos |
9 | 8 | sseld 3096 | . . . . . 6 tpos |
10 | elvvv 4602 | . . . . . 6 | |
11 | 9, 10 | syl6ib 160 | . . . . 5 tpos |
12 | 11 | pm4.71rd 391 | . . . 4 tpos tpos |
13 | 19.41vvv 1876 | . . . . 5 tpos tpos | |
14 | eleq1 2202 | . . . . . . . 8 tpos tpos | |
15 | df-br 3930 | . . . . . . . . 9 tpos tpos | |
16 | vex 2689 | . . . . . . . . . 10 | |
17 | vex 2689 | . . . . . . . . . 10 | |
18 | vex 2689 | . . . . . . . . . 10 | |
19 | brtposg 6151 | . . . . . . . . . 10 tpos | |
20 | 16, 17, 18, 19 | mp3an 1315 | . . . . . . . . 9 tpos |
21 | 15, 20 | bitr3i 185 | . . . . . . . 8 tpos |
22 | 14, 21 | syl6bb 195 | . . . . . . 7 tpos |
23 | 22 | pm5.32i 449 | . . . . . 6 tpos |
24 | 23 | 3exbii 1586 | . . . . 5 tpos |
25 | 13, 24 | bitr3i 185 | . . . 4 tpos |
26 | 12, 25 | syl6bb 195 | . . 3 tpos |
27 | 26 | abbi2dv 2258 | . 2 tpos |
28 | df-oprab 5778 | . 2 | |
29 | 27, 28 | syl6eqr 2190 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cab 2125 cvv 2686 wss 3071 cop 3530 class class class wbr 3929 cxp 4537 ccnv 4538 cdm 4539 wrel 4544 coprab 5775 tpos ctpos 6141 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-oprab 5778 df-tpos 6142 |
This theorem is referenced by: tposoprab 6177 |
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