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Mirrors > Home > ILE Home > Th. List > rntpos | Unicode version |
Description: The range of tpos when is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
rntpos | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2604 | . . . . 5 | |
2 | 1 | elrn 4595 | . . . 4 tpos tpos |
3 | vex 2604 | . . . . . . . . 9 | |
4 | 3, 1 | breldm 4557 | . . . . . . . 8 tpos tpos |
5 | dmtpos 5894 | . . . . . . . . 9 tpos | |
6 | 5 | eleq2d 2148 | . . . . . . . 8 tpos |
7 | 4, 6 | syl5ib 152 | . . . . . . 7 tpos |
8 | relcnv 4723 | . . . . . . . 8 | |
9 | elrel 4460 | . . . . . . . 8 | |
10 | 8, 9 | mpan 414 | . . . . . . 7 |
11 | 7, 10 | syl6 33 | . . . . . 6 tpos |
12 | breq1 3788 | . . . . . . . . 9 tpos tpos | |
13 | vex 2604 | . . . . . . . . . 10 | |
14 | vex 2604 | . . . . . . . . . 10 | |
15 | brtposg 5892 | . . . . . . . . . 10 tpos | |
16 | 13, 14, 1, 15 | mp3an 1268 | . . . . . . . . 9 tpos |
17 | 12, 16 | syl6bb 194 | . . . . . . . 8 tpos |
18 | 14, 13 | opex 3984 | . . . . . . . . 9 |
19 | 18, 1 | brelrn 4585 | . . . . . . . 8 |
20 | 17, 19 | syl6bi 161 | . . . . . . 7 tpos |
21 | 20 | exlimivv 1817 | . . . . . 6 tpos |
22 | 11, 21 | syli 37 | . . . . 5 tpos |
23 | 22 | exlimdv 1740 | . . . 4 tpos |
24 | 2, 23 | syl5bi 150 | . . 3 tpos |
25 | 1 | elrn 4595 | . . . 4 |
26 | 3, 1 | breldm 4557 | . . . . . . 7 |
27 | elrel 4460 | . . . . . . . 8 | |
28 | 27 | ex 113 | . . . . . . 7 |
29 | 26, 28 | syl5 32 | . . . . . 6 |
30 | breq1 3788 | . . . . . . . . 9 | |
31 | 30, 16 | syl6bbr 196 | . . . . . . . 8 tpos |
32 | 13, 14 | opex 3984 | . . . . . . . . 9 |
33 | 32, 1 | brelrn 4585 | . . . . . . . 8 tpos tpos |
34 | 31, 33 | syl6bi 161 | . . . . . . 7 tpos |
35 | 34 | exlimivv 1817 | . . . . . 6 tpos |
36 | 29, 35 | syli 37 | . . . . 5 tpos |
37 | 36 | exlimdv 1740 | . . . 4 tpos |
38 | 25, 37 | syl5bi 150 | . . 3 tpos |
39 | 24, 38 | impbid 127 | . 2 tpos |
40 | 39 | eqrdv 2079 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 103 wceq 1284 wex 1421 wcel 1433 cvv 2601 cop 3401 class class class wbr 3785 ccnv 4362 cdm 4363 crn 4364 wrel 4368 tpos ctpos 5882 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 df-tpos 5883 |
This theorem is referenced by: tposfo2 5905 |
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