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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 4606. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
dftpos3 | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4976 | . . . . . . . . . 10 | |
2 | dmtpos 6215 | . . . . . . . . . . 11 tpos | |
3 | 2 | releqd 4682 | . . . . . . . . . 10 tpos |
4 | 1, 3 | mpbiri 167 | . . . . . . . . 9 tpos |
5 | reltpos 6209 | . . . . . . . . 9 tpos | |
6 | 4, 5 | jctil 310 | . . . . . . . 8 tpos tpos |
7 | relrelss 5124 | . . . . . . . 8 tpos tpos tpos | |
8 | 6, 7 | sylib 121 | . . . . . . 7 tpos |
9 | 8 | sseld 3136 | . . . . . 6 tpos |
10 | elvvv 4661 | . . . . . 6 | |
11 | 9, 10 | syl6ib 160 | . . . . 5 tpos |
12 | 11 | pm4.71rd 392 | . . . 4 tpos tpos |
13 | 19.41vvv 1891 | . . . . 5 tpos tpos | |
14 | eleq1 2227 | . . . . . . . 8 tpos tpos | |
15 | df-br 3977 | . . . . . . . . 9 tpos tpos | |
16 | vex 2724 | . . . . . . . . . 10 | |
17 | vex 2724 | . . . . . . . . . 10 | |
18 | vex 2724 | . . . . . . . . . 10 | |
19 | brtposg 6213 | . . . . . . . . . 10 tpos | |
20 | 16, 17, 18, 19 | mp3an 1326 | . . . . . . . . 9 tpos |
21 | 15, 20 | bitr3i 185 | . . . . . . . 8 tpos |
22 | 14, 21 | bitrdi 195 | . . . . . . 7 tpos |
23 | 22 | pm5.32i 450 | . . . . . 6 tpos |
24 | 23 | 3exbii 1594 | . . . . 5 tpos |
25 | 13, 24 | bitr3i 185 | . . . 4 tpos |
26 | 12, 25 | bitrdi 195 | . . 3 tpos |
27 | 26 | abbi2dv 2283 | . 2 tpos |
28 | df-oprab 5840 | . 2 | |
29 | 27, 28 | eqtr4di 2215 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cab 2150 cvv 2721 wss 3111 cop 3573 class class class wbr 3976 cxp 4596 ccnv 4597 cdm 4598 wrel 4603 coprab 5837 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-oprab 5840 df-tpos 6204 |
This theorem is referenced by: tposoprab 6239 |
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