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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 17770 | . . . 4 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))) |
8 | 7 | breqd 5163 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))𝐺)) |
9 | brin 5204 | . . 3 ⊢ (𝐹((𝑋𝑆𝑌) ∩ ◡(𝑌𝑆𝑋))𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐹◡(𝑌𝑆𝑋)𝐺)) | |
10 | 8, 9 | bitrdi 286 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐹◡(𝑌𝑆𝑋)𝐺))) |
11 | eqid 2725 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqid 2725 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | eqid 2725 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
14 | 1, 11, 12, 13, 6, 3, 5, 4 | sectss 17763 | . . . . 5 ⊢ (𝜑 → (𝑌𝑆𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
15 | relxp 5699 | . . . . 5 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
16 | relss 5786 | . . . . 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 6114 | . . . 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 628 | . 2 ⊢ (𝜑 → ((𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐹◡(𝑌𝑆𝑋)𝐺) ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹))) |
21 | 10, 20 | bitrd 278 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3945 ⊆ wss 3946 class class class wbr 5152 × cxp 5679 ◡ccnv 5680 Rel wrel 5686 ‘cfv 6553 (class class class)co 7423 Basecbs 17208 Hom chom 17272 compcco 17273 Catccat 17672 Idccid 17673 Sectcsect 17755 Invcinv 17756 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 8002 df-2nd 8003 df-sect 17758 df-inv 17759 |
This theorem is referenced by: invsym 17773 invfun 17775 invco 17782 inveq 17785 monsect 17794 invid 17798 invcoisoid 17803 isocoinvid 17804 funcinv 17887 fthinv 17943 fucinv 17993 invfuc 17994 2initoinv 18027 2termoinv 18034 setcinv 18107 catcisolem 18127 catciso 18128 rngcinv 20610 ringcinv 20644 rngcinvALTV 47590 ringcinvALTV 47624 thincinv 48317 |
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