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Mirrors > Home > MPE Home > Th. List > isinv | Structured version Visualization version GIF version |
Description: Value of the inverse relation. (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΅) |
invfval.s | β’ π = (SectβπΆ) |
Ref | Expression |
---|---|
isinv | β’ (π β (πΉ(πππ)πΊ β (πΉ(πππ)πΊ β§ πΊ(πππ)πΉ))) |
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.x | . . . . 5 β’ (π β π β π΅) | |
5 | invfval.y | . . . . 5 β’ (π β π β π΅) | |
6 | invfval.s | . . . . 5 β’ π = (SectβπΆ) | |
7 | 1, 2, 3, 4, 5, 6 | invfval 17706 | . . . 4 β’ (π β (πππ) = ((πππ) β© β‘(πππ))) |
8 | 7 | breqd 5160 | . . 3 β’ (π β (πΉ(πππ)πΊ β πΉ((πππ) β© β‘(πππ))πΊ)) |
9 | brin 5201 | . . 3 β’ (πΉ((πππ) β© β‘(πππ))πΊ β (πΉ(πππ)πΊ β§ πΉβ‘(πππ)πΊ)) | |
10 | 8, 9 | bitrdi 287 | . 2 β’ (π β (πΉ(πππ)πΊ β (πΉ(πππ)πΊ β§ πΉβ‘(πππ)πΊ))) |
11 | eqid 2733 | . . . . . 6 β’ (Hom βπΆ) = (Hom βπΆ) | |
12 | eqid 2733 | . . . . . 6 β’ (compβπΆ) = (compβπΆ) | |
13 | eqid 2733 | . . . . . 6 β’ (IdβπΆ) = (IdβπΆ) | |
14 | 1, 11, 12, 13, 6, 3, 5, 4 | sectss 17699 | . . . . 5 β’ (π β (πππ) β ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π))) |
15 | relxp 5695 | . . . . 5 β’ Rel ((π(Hom βπΆ)π) Γ (π(Hom βπΆ)π)) | |
16 | relss 5782 | . . . . 5 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))) | |
17 | 14, 15, 16 | mpisyl 21 | . . . 4 β’ (π β Rel (πππ)) |
18 | relbrcnvg 6105 | . . . 4 β’ (Rel (πππ) β (πΉβ‘(πππ)πΊ β πΊ(πππ)πΉ)) | |
19 | 17, 18 | syl 17 | . . 3 β’ (π β (πΉβ‘(πππ)πΊ β πΊ(πππ)πΉ)) |
20 | 19 | anbi2d 630 | . 2 β’ (π β ((πΉ(πππ)πΊ β§ πΉβ‘(πππ)πΊ) β (πΉ(πππ)πΊ β§ πΊ(πππ)πΉ))) |
21 | 10, 20 | bitrd 279 | 1 β’ (π β (πΉ(πππ)πΊ β (πΉ(πππ)πΊ β§ πΊ(πππ)πΉ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β© cin 3948 β wss 3949 class class class wbr 5149 Γ cxp 5675 β‘ccnv 5676 Rel wrel 5682 βcfv 6544 (class class class)co 7409 Basecbs 17144 Hom chom 17208 compcco 17209 Catccat 17608 Idccid 17609 Sectcsect 17691 Invcinv 17692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-sect 17694 df-inv 17695 |
This theorem is referenced by: invsym 17709 invfun 17711 invco 17718 inveq 17721 monsect 17730 invid 17734 invcoisoid 17739 isocoinvid 17740 funcinv 17823 fthinv 17877 fucinv 17926 invfuc 17927 2initoinv 17960 2termoinv 17967 setcinv 18040 catcisolem 18060 catciso 18061 rngcinv 46879 rngcinvALTV 46891 ringcinv 46930 ringcinvALTV 46954 thincinv 47679 |
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