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Mirrors > Home > MPE Home > Th. List > tposf12 | Structured version Visualization version Unicode version |
Description: Condition for an injective transposition. (Contributed by NM, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposf12 | tpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 477 | . . . 4 | |
2 | relcnv 5503 | . . . . . . 7 | |
3 | cnvf1o 7276 | . . . . . . 7 | |
4 | f1of1 6136 | . . . . . . 7 | |
5 | 2, 3, 4 | mp2b 10 | . . . . . 6 |
6 | simpl 473 | . . . . . . . 8 | |
7 | dfrel2 5583 | . . . . . . . 8 | |
8 | 6, 7 | sylib 208 | . . . . . . 7 |
9 | f1eq3 6098 | . . . . . . 7 | |
10 | 8, 9 | syl 17 | . . . . . 6 |
11 | 5, 10 | mpbii 223 | . . . . 5 |
12 | f1dm 6105 | . . . . . . . 8 | |
13 | 1, 12 | syl 17 | . . . . . . 7 |
14 | 13 | cnveqd 5298 | . . . . . 6 |
15 | mpteq1 4737 | . . . . . 6 | |
16 | f1eq1 6096 | . . . . . 6 | |
17 | 14, 15, 16 | 3syl 18 | . . . . 5 |
18 | 11, 17 | mpbird 247 | . . . 4 |
19 | f1co 6110 | . . . 4 | |
20 | 1, 18, 19 | syl2anc 693 | . . 3 |
21 | 12 | releqd 5203 | . . . . 5 |
22 | 21 | biimparc 504 | . . . 4 |
23 | dftpos2 7369 | . . . 4 tpos | |
24 | f1eq1 6096 | . . . 4 tpos tpos | |
25 | 22, 23, 24 | 3syl 18 | . . 3 tpos |
26 | 20, 25 | mpbird 247 | . 2 tpos |
27 | 26 | ex 450 | 1 tpos |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 csn 4177 cuni 4436 cmpt 4729 ccnv 5113 cdm 5114 ccom 5118 wrel 5119 wf1 5885 wf1o 5887 tpos ctpos 7351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-1st 7168 df-2nd 7169 df-tpos 7352 |
This theorem is referenced by: tposf1o2 7378 |
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