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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 4200 | . . 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 2732 | . . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
8 | eqid 2732 | . . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂) | |
9 | 2, 3, 4, 5, 6, 7, 8 | oppcsect2 17722 | . . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋)) |
10 | 9 | cnveqd 5873 | . . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋)) |
11 | eqid 2732 | . . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqid 2732 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | eqid 2732 | . . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
14 | 2, 11, 12, 13, 7, 4, 5, 6 | sectss 17695 | . . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
15 | relxp 5693 | . . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
16 | relss 5779 | . . . . . . 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 6185 | . . . . . 6 ⊢ (Rel (𝑌(Sect‘𝐶)𝑋) ↔ ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋)) | |
19 | 17, 18 | sylib 217 | . . . . 5 ⊢ (𝜑 → ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋)) |
20 | 10, 19 | eqtrd 2772 | . . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋)) |
21 | 2, 3, 4, 6, 5, 7, 8 | oppcsect2 17722 | . . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌)) |
22 | 20, 21 | ineq12d 4212 | . . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
23 | 1, 22 | eqtrid 2784 | . 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
24 | 3, 2 | oppcbas 17659 | . . 3 ⊢ 𝐵 = (Base‘𝑂) |
25 | oppcinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝑂) | |
26 | 3 | oppccat 17664 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
27 | 4, 26 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
28 | 24, 25, 27, 6, 5, 8 | invfval 17702 | . 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 17702 | . 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
31 | 23, 28, 30 | 3eqtr4d 2782 | 1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∩ cin 3946 ⊆ wss 3947 × cxp 5673 ◡ccnv 5674 Rel wrel 5680 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 Hom chom 17204 compcco 17205 Catccat 17604 Idccid 17605 oppCatcoppc 17651 Sectcsect 17687 Invcinv 17688 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-hom 17217 df-cco 17218 df-cat 17608 df-cid 17609 df-oppc 17652 df-sect 17690 df-inv 17691 |
This theorem is referenced by: oppciso 17724 episect 17728 |
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