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Mirrors > Home > MPE Home > Th. List > invss | Structured version Visualization version GIF version |
Description: The inverse relation is a relation between morphisms 𝐹:𝑋⟶𝑌 and their inverses 𝐺:𝑌⟶𝑋. (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) |
invss.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
invss | ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invfval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invfval.n | . . . 4 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | invfval.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | invfval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | invfval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | eqid 2622 | . . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
7 | 1, 2, 3, 4, 5, 6 | invfval 16419 | . . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) = ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋))) |
8 | inss1 3833 | . . 3 ⊢ ((𝑋(Sect‘𝐶)𝑌) ∩ ◡(𝑌(Sect‘𝐶)𝑋)) ⊆ (𝑋(Sect‘𝐶)𝑌) | |
9 | 7, 8 | syl6eqss 3655 | . 2 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ (𝑋(Sect‘𝐶)𝑌)) |
10 | invss.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
11 | eqid 2622 | . . 3 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
12 | eqid 2622 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
13 | 1, 10, 11, 12, 6, 3, 4, 5 | sectss 16412 | . 2 ⊢ (𝜑 → (𝑋(Sect‘𝐶)𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
14 | 9, 13 | sstrd 3613 | 1 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 × cxp 5112 ◡ccnv 5113 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Hom chom 15952 compcco 15953 Catccat 16325 Idccid 16326 Sectcsect 16404 Invcinv 16405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-sect 16407 df-inv 16408 |
This theorem is referenced by: invsym2 16423 invfun 16424 isohom 16436 invfuc 16634 |
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