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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 2725 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | eqid 2725 | . . 3 ⊢ (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) = (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) | |
10 | eqid 2725 | . . 3 ⊢ (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) = (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | dfiso2 17758 | . 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 2725 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | dfiso3.s | . . . . . 6 ⊢ 𝑆 = (Sect‘𝐶) | |
14 | 3 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat) |
15 | 6 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌 ∈ 𝐵) |
16 | 5 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋 ∈ 𝐵) |
17 | simpr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋)) | |
18 | 7 | adantr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌)) |
19 | 1, 2, 12, 8, 13, 14, 15, 16, 17, 18 | issect2 17740 | . . . . 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 17740 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))) |
21 | 19, 20 | anbi12d 630 | . . . 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 459 | . . . 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 | bitr2di 287 | . . 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 3166 | . 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 278 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 〈cop 4636 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Hom chom 17247 compcco 17248 Catccat 17647 Idccid 17648 Sectcsect 17730 Isociso 17732 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-sect 17733 df-inv 17734 df-iso 17735 |
This theorem is referenced by: thinciso 48252 |
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