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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 4976 | . . . . . . 7 | |
3 | cnvf1o 6184 | . . . . . . 7 | |
4 | f1of1 5425 | . . . . . . 7 | |
5 | 2, 3, 4 | mp2b 8 | . . . . . 6 |
6 | simpl 108 | . . . . . . . 8 | |
7 | dfrel2 5048 | . . . . . . . 8 | |
8 | 6, 7 | sylib 121 | . . . . . . 7 |
9 | f1eq3 5384 | . . . . . . 7 | |
10 | 8, 9 | syl 14 | . . . . . 6 |
11 | 5, 10 | mpbii 147 | . . . . 5 |
12 | f1dm 5392 | . . . . . . . 8 | |
13 | 1, 12 | syl 14 | . . . . . . 7 |
14 | 13 | cnveqd 4774 | . . . . . 6 |
15 | mpteq1 4060 | . . . . . 6 | |
16 | f1eq1 5382 | . . . . . 6 | |
17 | 14, 15, 16 | 3syl 17 | . . . . 5 |
18 | 11, 17 | mpbird 166 | . . . 4 |
19 | f1co 5399 | . . . 4 | |
20 | 1, 18, 19 | syl2anc 409 | . . 3 |
21 | 12 | releqd 4682 | . . . . 5 |
22 | 21 | biimparc 297 | . . . 4 |
23 | dftpos2 6220 | . . . 4 tpos | |
24 | f1eq1 5382 | . . . 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 1342 csn 3570 cuni 3783 cmpt 4037 ccnv 4597 cdm 4598 ccom 4602 wrel 4603 wf1 5179 wf1o 5181 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-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-tpos 6204 |
This theorem is referenced by: tposf1o2 6229 |
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