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Mirrors > Home > MPE Home > Th. List > oppcinv | Structured version Visualization version GIF version |
Description: An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcsect.b | ⊢ 𝐵 = (Base‘𝐶) |
oppcsect.o | ⊢ 𝑂 = (oppCat‘𝐶) |
oppcsect.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
oppcsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
oppcsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
oppcinv.s | ⊢ 𝐼 = (Inv‘𝐶) |
oppcinv.t | ⊢ 𝐽 = (Inv‘𝑂) |
Ref | Expression |
---|---|
oppcinv | ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4091 | . . 3 ⊢ ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) | |
2 | oppcsect.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
3 | oppcsect.o | . . . . . . 7 ⊢ 𝑂 = (oppCat‘𝐶) | |
4 | oppcsect.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | oppcsect.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | oppcsect.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | eqid 2738 | . . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
8 | eqid 2738 | . . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂) | |
9 | 2, 3, 4, 5, 6, 7, 8 | oppcsect2 17154 | . . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋)) |
10 | 9 | cnveqd 5718 | . . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋)) |
11 | eqid 2738 | . . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqid 2738 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | eqid 2738 | . . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
14 | 2, 11, 12, 13, 7, 4, 5, 6 | sectss 17127 | . . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
15 | relxp 5543 | . . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
16 | relss 5627 | . . . . . . 7 ⊢ ((𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌(Sect‘𝐶)𝑋))) | |
17 | 14, 15, 16 | mpisyl 21 | . . . . . 6 ⊢ (𝜑 → Rel (𝑌(Sect‘𝐶)𝑋)) |
18 | dfrel2 6021 | . . . . . 6 ⊢ (Rel (𝑌(Sect‘𝐶)𝑋) ↔ ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋)) | |
19 | 17, 18 | sylib 221 | . . . . 5 ⊢ (𝜑 → ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋)) |
20 | 10, 19 | eqtrd 2773 | . . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋)) |
21 | 2, 3, 4, 6, 5, 7, 8 | oppcsect2 17154 | . . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌)) |
22 | 20, 21 | ineq12d 4104 | . . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
23 | 1, 22 | syl5eq 2785 | . 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
24 | 3, 2 | oppcbas 17092 | . . 3 ⊢ 𝐵 = (Base‘𝑂) |
25 | oppcinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝑂) | |
26 | 3 | oppccat 17096 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
27 | 4, 26 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
28 | 24, 25, 27, 6, 5, 8 | invfval 17134 | . 2 ⊢ (𝜑 → (𝑋𝐽𝑌) = ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋))) |
29 | oppcinv.s | . . 3 ⊢ 𝐼 = (Inv‘𝐶) | |
30 | 2, 29, 4, 5, 6, 7 | invfval 17134 | . 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
31 | 23, 28, 30 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 ⊆ wss 3843 × cxp 5523 ◡ccnv 5524 Rel wrel 5530 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 Hom chom 16679 compcco 16680 Catccat 17038 Idccid 17039 oppCatcoppc 17085 Sectcsect 17119 Invcinv 17120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-hom 16692 df-cco 16693 df-cat 17042 df-cid 17043 df-oppc 17086 df-sect 17122 df-inv 17123 |
This theorem is referenced by: oppciso 17156 episect 17160 |
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