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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 2825 | . . 3 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | eqid 2825 | . . 3 ⊢ (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) = (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) | |
10 | eqid 2825 | . . 3 ⊢ (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) = (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | dfiso2 16784 | . 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 2825 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | dfiso3.s | . . . . . 6 ⊢ 𝑆 = (Sect‘𝐶) | |
14 | 3 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐶 ∈ Cat) |
15 | 6 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑌 ∈ 𝐵) |
16 | 5 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑋 ∈ 𝐵) |
17 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝑔 ∈ (𝑌𝐻𝑋)) | |
18 | 7 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → 𝐹 ∈ (𝑋𝐻𝑌)) |
19 | 1, 2, 12, 8, 13, 14, 15, 16, 17, 18 | issect2 16766 | . . . . 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 16766 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑌𝐻𝑋)) → (𝐹(𝑋𝑆𝑌)𝑔 ↔ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))) |
21 | 19, 20 | anbi12d 626 | . . . 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 454 | . . . 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 280 | . . 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 3259 | . 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 271 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ ∃𝑔 ∈ (𝑌𝐻𝑋)(𝑔(𝑌𝑆𝑋)𝐹 ∧ 𝐹(𝑋𝑆𝑌)𝑔))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3118 〈cop 4403 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 Hom chom 16316 compcco 16317 Catccat 16677 Idccid 16678 Sectcsect 16756 Isociso 16758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-sect 16759 df-inv 16760 df-iso 16761 |
This theorem is referenced by: (None) |
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