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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 109 | . . . 4 | |
2 | relcnv 4912 | . . . . . . 7 | |
3 | cnvf1o 6115 | . . . . . . 7 | |
4 | f1of1 5359 | . . . . . . 7 | |
5 | 2, 3, 4 | mp2b 8 | . . . . . 6 |
6 | simpl 108 | . . . . . . . 8 | |
7 | dfrel2 4984 | . . . . . . . 8 | |
8 | 6, 7 | sylib 121 | . . . . . . 7 |
9 | f1eq3 5320 | . . . . . . 7 | |
10 | 8, 9 | syl 14 | . . . . . 6 |
11 | 5, 10 | mpbii 147 | . . . . 5 |
12 | f1dm 5328 | . . . . . . . 8 | |
13 | 1, 12 | syl 14 | . . . . . . 7 |
14 | 13 | cnveqd 4710 | . . . . . 6 |
15 | mpteq1 4007 | . . . . . 6 | |
16 | f1eq1 5318 | . . . . . 6 | |
17 | 14, 15, 16 | 3syl 17 | . . . . 5 |
18 | 11, 17 | mpbird 166 | . . . 4 |
19 | f1co 5335 | . . . 4 | |
20 | 1, 18, 19 | syl2anc 408 | . . 3 |
21 | 12 | releqd 4618 | . . . . 5 |
22 | 21 | biimparc 297 | . . . 4 |
23 | dftpos2 6151 | . . . 4 tpos | |
24 | f1eq1 5318 | . . . 4 tpos tpos | |
25 | 22, 23, 24 | 3syl 17 | . . 3 tpos |
26 | 20, 25 | mpbird 166 | . 2 tpos |
27 | 26 | ex 114 | 1 tpos |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 csn 3522 cuni 3731 cmpt 3984 ccnv 4533 cdm 4534 ccom 4538 wrel 4539 wf1 5115 wf1o 5117 tpos ctpos 6134 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-1st 6031 df-2nd 6032 df-tpos 6135 |
This theorem is referenced by: tposf1o2 6160 |
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