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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 | ⊢ 𝐵 = (Base‘𝐶) |
invfval.n | ⊢ 𝑁 = (Inv‘𝐶) |
invfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
invfval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
invfval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
invfval.s | ⊢ 𝑆 = (Sect‘𝐶) |
Ref | Expression |
---|---|
isinv | ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . . . . 5 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | invfval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | invfval.s | . . . . 5 ⊢ 𝑆 = (Sect‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | invfval 17021 | . . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) |
8 | 7 | breqd 5041 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))𝐺)) |
9 | brin 5082 | . . 3 ⊢ (𝐹((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐹◡(𝑌𝑆𝑋)𝐺)) | |
10 | 8, 9 | syl6bb 290 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐹◡(𝑌𝑆𝑋)𝐺))) |
11 | eqid 2798 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqid 2798 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | eqid 2798 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
14 | 1, 11, 12, 13, 6, 3, 5, 4 | sectss 17014 | . . . . 5 ⊢ (𝜑 → (𝑌𝑆𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
15 | relxp 5537 | . . . . 5 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
16 | relss 5620 | . . . . 5 ⊢ ((𝑌𝑆𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌𝑆𝑋))) | |
17 | 14, 15, 16 | mpisyl 21 | . . . 4 ⊢ (𝜑 → Rel (𝑌𝑆𝑋)) |
18 | relbrcnvg 5935 | . . . 4 ⊢ (Rel (𝑌𝑆𝑋) → (𝐹◡(𝑌𝑆𝑋)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹)) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → (𝐹◡(𝑌𝑆𝑋)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹)) |
20 | 19 | anbi2d 631 | . 2 ⊢ (𝜑 → ((𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐹◡(𝑌𝑆𝑋)𝐺) ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹))) |
21 | 10, 20 | bitrd 282 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 × cxp 5517 ◡ccnv 5518 Rel wrel 5524 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 Idccid 16928 Sectcsect 17006 Invcinv 17007 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-sect 17009 df-inv 17010 |
This theorem is referenced by: invsym 17024 invfun 17026 invco 17033 inveq 17036 monsect 17045 invid 17049 invcoisoid 17054 isocoinvid 17055 funcinv 17135 fthinv 17188 fucinv 17235 invfuc 17236 2initoinv 17262 2termoinv 17269 setcinv 17342 catcisolem 17358 catciso 17359 rngcinv 44605 rngcinvALTV 44617 ringcinv 44656 ringcinvALTV 44680 |
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