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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincinv | Structured version Visualization version GIF version |
Description: In a thin category, πΉ is an inverse of πΊ iff πΉ is a section of πΊ (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
thincsect.c | β’ (π β πΆ β ThinCat) |
thincsect.b | β’ π΅ = (BaseβπΆ) |
thincsect.x | β’ (π β π β π΅) |
thincsect.y | β’ (π β π β π΅) |
thincsect.s | β’ π = (SectβπΆ) |
thincinv.n | β’ π = (InvβπΆ) |
Ref | Expression |
---|---|
thincinv | β’ (π β (πΉ(πππ)πΊ β πΉ(πππ)πΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thincsect.b | . . 3 β’ π΅ = (BaseβπΆ) | |
2 | thincinv.n | . . 3 β’ π = (InvβπΆ) | |
3 | thincsect.c | . . . 4 β’ (π β πΆ β ThinCat) | |
4 | 3 | thinccd 47198 | . . 3 β’ (π β πΆ β Cat) |
5 | thincsect.x | . . 3 β’ (π β π β π΅) | |
6 | thincsect.y | . . 3 β’ (π β π β π΅) | |
7 | thincsect.s | . . 3 β’ π = (SectβπΆ) | |
8 | 1, 2, 4, 5, 6, 7 | isinv 17672 | . 2 β’ (π β (πΉ(πππ)πΊ β (πΉ(πππ)πΊ β§ πΊ(πππ)πΉ))) |
9 | 3, 1, 5, 6, 7 | thincsect2 47231 | . . 3 β’ (π β (πΉ(πππ)πΊ β πΊ(πππ)πΉ)) |
10 | 9 | biimpa 477 | . 2 β’ ((π β§ πΉ(πππ)πΊ) β πΊ(πππ)πΉ) |
11 | 8, 10 | mpbiran3d 47035 | 1 β’ (π β (πΉ(πππ)πΊ β πΉ(πππ)πΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1541 β wcel 2106 class class class wbr 5125 βcfv 6516 (class class class)co 7377 Basecbs 17109 Sectcsect 17656 Invcinv 17657 ThinCatcthinc 47192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-id 5551 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-1st 7941 df-2nd 7942 df-cat 17577 df-cid 17578 df-sect 17659 df-inv 17660 df-thinc 47193 |
This theorem is referenced by: (None) |
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