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Mirrors > Home > ILE Home > Th. List > tposf12 | Unicode version |
Description: Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposf12 | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 103 | . . . 4 | |
2 | relcnv 4703 | . . . . . . 7 | |
3 | cnvf1o 5846 | . . . . . . 7 | |
4 | f1of1 5125 | . . . . . . 7 | |
5 | 2, 3, 4 | mp2b 8 | . . . . . 6 |
6 | simpl 102 | . . . . . . . 8 | |
7 | dfrel2 4771 | . . . . . . . 8 | |
8 | 6, 7 | sylib 127 | . . . . . . 7 |
9 | f1eq3 5089 | . . . . . . 7 | |
10 | 8, 9 | syl 14 | . . . . . 6 |
11 | 5, 10 | mpbii 136 | . . . . 5 |
12 | f1dm 5096 | . . . . . . . 8 | |
13 | 1, 12 | syl 14 | . . . . . . 7 |
14 | 13 | cnveqd 4511 | . . . . . 6 |
15 | mpteq1 3841 | . . . . . 6 | |
16 | f1eq1 5087 | . . . . . 6 | |
17 | 14, 15, 16 | 3syl 17 | . . . . 5 |
18 | 11, 17 | mpbird 156 | . . . 4 |
19 | f1co 5101 | . . . 4 | |
20 | 1, 18, 19 | syl2anc 391 | . . 3 |
21 | 12 | releqd 4424 | . . . . 5 |
22 | 21 | biimparc 283 | . . . 4 |
23 | dftpos2 5876 | . . . 4 tpos | |
24 | f1eq1 5087 | . . . 4 tpos tpos | |
25 | 22, 23, 24 | 3syl 17 | . . 3 tpos |
26 | 20, 25 | mpbird 156 | . 2 tpos |
27 | 26 | ex 108 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 csn 3375 cuni 3580 cmpt 3818 ccnv 4344 cdm 4345 ccom 4349 wrel 4350 wf1 4899 wf1o 4901 tpos ctpos 5859 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-1st 5767 df-2nd 5768 df-tpos 5860 |
This theorem is referenced by: tposf1o2 5885 |
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