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| Mirrors > Home > MPE Home > Th. List > Mathboxes > thinccic | Structured version Visualization version GIF version | ||
| Description: In a thin category, two objects are isomorphic iff there are morphisms between them in both directions. (Contributed by Zhi Wang, 25-Sep-2024.) |
| Ref | Expression |
|---|---|
| thincsect.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
| thincsect.b | ⊢ 𝐵 = (Base‘𝐶) |
| thincsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| thincsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| thinciso.h | ⊢ 𝐻 = (Hom ‘𝐶) |
| Ref | Expression |
|---|---|
| thinccic | ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | thincsect.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | thinciso.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
| 3 | eqid 2737 | . . . . . 6 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
| 4 | thincsect.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
| 5 | 4 | thinccd 49073 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) |
| 6 | thincsect.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 7 | thincsect.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 8 | 1, 2, 3, 5, 6, 7 | isohom 17820 | . . . . 5 ⊢ (𝜑 → (𝑋(Iso‘𝐶)𝑌) ⊆ (𝑋𝐻𝑌)) |
| 9 | 8 | sselda 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌)) |
| 10 | 4 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat) |
| 11 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵) |
| 12 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵) |
| 13 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌)) | |
| 14 | 10, 1, 11, 12, 2, 3, 13 | thinciso 49117 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑓 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (𝑌𝐻𝑋) ≠ ∅)) |
| 15 | 9, 14 | biadanid 823 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅))) |
| 16 | 15 | exbidv 1921 | . 2 ⊢ (𝜑 → (∃𝑓 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ ∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅))) |
| 17 | 3, 1, 5, 6, 7 | cic 17843 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌))) |
| 18 | n0 4353 | . . . . 5 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
| 19 | 18 | anbi1i 624 | . . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅)) |
| 20 | 19.41v 1949 | . . . 4 ⊢ (∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅)) | |
| 21 | 19, 20 | bitr4i 278 | . . 3 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ ∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅)) |
| 22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ ∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅))) |
| 23 | 16, 17, 22 | 3bitr4d 311 | 1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 Isociso 17790 ≃𝑐 ccic 17839 ThinCatcthinc 49067 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-supp 8186 df-cat 17711 df-cid 17712 df-sect 17791 df-inv 17792 df-iso 17793 df-cic 17840 df-thinc 49068 |
| This theorem is referenced by: postc 49173 |
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