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Mirrors > Home > MPE Home > Th. List > dfiso3 | Structured version Visualization version GIF version |
Description: Alternate definition of an isomorphism of a category as a section in both directions. (Contributed by AV, 11-Apr-2020.) |
Ref | Expression |
---|---|
dfiso3.b | ⊢ 𝐵 = (Base‘𝐶) |
dfiso3.h | ⊢ 𝐻 = (Hom ‘𝐶) |
dfiso3.i | ⊢ 𝐼 = (Iso‘𝐶) |
dfiso3.s | ⊢ 𝑆 = (Sect‘𝐶) |
dfiso3.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
dfiso3.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
dfiso3.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
dfiso3.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
dfiso3 | ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiso3.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | dfiso3.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | dfiso3.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | dfiso3.i | . . 3 ⊢ 𝐼 = (Iso‘𝐶) | |
5 | dfiso3.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | dfiso3.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | dfiso3.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
8 | eqid 2798 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | eqid 2798 | . . 3 ⊢ (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) = (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) | |
10 | eqid 2798 | . . 3 ⊢ (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) = (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | dfiso2 17034 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)))) |
12 | eqid 2798 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | dfiso3.s | . . . . . 6 ⊢ 𝑆 = (Sect‘𝐶) | |
14 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat) |
15 | 6 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌 ∈ 𝐵) |
16 | 5 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋 ∈ 𝐵) |
17 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋)) | |
18 | 7 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌)) |
19 | 1, 2, 12, 8, 13, 14, 15, 16, 17, 18 | issect2 17016 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝑔(𝑌𝑆𝑋)𝐹 ↔ (𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌))) |
20 | 1, 2, 12, 8, 13, 14, 16, 15, 18, 17 | issect2 17016 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))) |
21 | 19, 20 | anbi12d 633 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → ((𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔) ↔ ((𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌) ∧ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))) |
22 | ancom 464 | . . . 4 ⊢ (((𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌) ∧ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌))) | |
23 | 21, 22 | syl6rbb 291 | . . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (((𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)) ↔ (𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔))) |
24 | 23 | rexbidva 3255 | . 2 ⊢ (𝜑 → (∃𝑔 ∈ (𝑌𝐻𝑋)((𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ∧ (𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝑔) = ((Id‘𝐶)‘𝑌)) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔))) |
25 | 11, 24 | bitrd 282 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 〈cop 4531 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 Idccid 16928 Sectcsect 17006 Isociso 17008 |
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 df-iso 17011 |
This theorem is referenced by: (None) |
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