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Mirrors > Home > MPE Home > Th. List > invsym2 | Structured version Visualization version GIF version |
Description: The inverse relation is symmetric. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
invfval.b | β’ π΅ = (BaseβπΆ) |
invfval.n | β’ π = (InvβπΆ) |
invfval.c | β’ (π β πΆ β Cat) |
invfval.x | β’ (π β π β π΅) |
invfval.y | β’ (π β π β π΅) |
Ref | Expression |
---|---|
invsym2 | β’ (π β β‘(πππ) = (πππ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . . 5 β’ π΅ = (BaseβπΆ) | |
2 | invfval.n | . . . . 5 β’ π = (InvβπΆ) | |
3 | invfval.c | . . . . 5 β’ (π β πΆ β Cat) | |
4 | invfval.y | . . . . 5 β’ (π β π β π΅) | |
5 | invfval.x | . . . . 5 β’ (π β π β π΅) | |
6 | eqid 2728 | . . . . 5 β’ (Hom βπΆ) = (Hom βπΆ) | |
7 | 1, 2, 3, 4, 5, 6 | invss 17737 | . . . 4 β’ (π β (πππ) β ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π))) |
8 | relxp 5690 | . . . 4 β’ Rel ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π)) | |
9 | relss 5777 | . . . 4 β’ ((πππ) β ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π)) β (Rel ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π)) β Rel (πππ))) | |
10 | 7, 8, 9 | mpisyl 21 | . . 3 β’ (π β Rel (πππ)) |
11 | relcnv 6102 | . . 3 β’ Rel β‘(πππ) | |
12 | 10, 11 | jctil 519 | . 2 β’ (π β (Rel β‘(πππ) β§ Rel (πππ))) |
13 | 1, 2, 3, 5, 4 | invsym 17738 | . . . 4 β’ (π β (π(πππ)π β π(πππ)π)) |
14 | vex 3474 | . . . . . 6 β’ π β V | |
15 | vex 3474 | . . . . . 6 β’ π β V | |
16 | 14, 15 | brcnv 5879 | . . . . 5 β’ (πβ‘(πππ)π β π(πππ)π) |
17 | df-br 5143 | . . . . 5 β’ (πβ‘(πππ)π β β¨π, πβ© β β‘(πππ)) | |
18 | 16, 17 | bitr3i 277 | . . . 4 β’ (π(πππ)π β β¨π, πβ© β β‘(πππ)) |
19 | df-br 5143 | . . . 4 β’ (π(πππ)π β β¨π, πβ© β (πππ)) | |
20 | 13, 18, 19 | 3bitr3g 313 | . . 3 β’ (π β (β¨π, πβ© β β‘(πππ) β β¨π, πβ© β (πππ))) |
21 | 20 | eqrelrdv2 5791 | . 2 β’ (((Rel β‘(πππ) β§ Rel (πππ)) β§ π) β β‘(πππ) = (πππ)) |
22 | 12, 21 | mpancom 687 | 1 β’ (π β β‘(πππ) = (πππ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3945 β¨cop 4630 class class class wbr 5142 Γ cxp 5670 β‘ccnv 5671 Rel wrel 5677 βcfv 6542 (class class class)co 7414 Basecbs 17173 Hom chom 17237 Catccat 17637 Invcinv 17721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-sect 17723 df-inv 17724 |
This theorem is referenced by: invf 17744 invf1o 17745 invinv 17746 cicsym 17780 |
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