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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 2821 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | eqid 2821 | . . 3 ⊢ (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) = (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) | |
10 | eqid 2821 | . . 3 ⊢ (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) = (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | dfiso2 17036 | . 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 2821 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | dfiso3.s | . . . . . 6 ⊢ 𝑆 = (Sect‘𝐶) | |
14 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat) |
15 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌 ∈ 𝐵) |
16 | 5 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋 ∈ 𝐵) |
17 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋)) | |
18 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌)) |
19 | 1, 2, 12, 8, 13, 14, 15, 16, 17, 18 | issect2 17018 | . . . . 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 17018 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))) |
21 | 19, 20 | anbi12d 632 | . . . 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 463 | . . . 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 290 | . . 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 3296 | . 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 281 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 〈cop 4566 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 compcco 16571 Catccat 16929 Idccid 16930 Sectcsect 17008 Isociso 17010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-sect 17011 df-inv 17012 df-iso 17013 |
This theorem is referenced by: (None) |
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