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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 2736 | . . . . . 6 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
4 | thincsect.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
5 | 4 | thinccd 45922 | . . . . . 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 17235 | . . . . 5 ⊢ (𝜑 → (𝑋(Iso‘𝐶)𝑌) ⊆ (𝑋𝐻𝑌)) |
9 | 8 | sselda 3887 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌)) |
10 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat) |
11 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵) |
12 | 7 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵) |
13 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌)) | |
14 | 10, 1, 11, 12, 2, 3, 13 | thinciso 45957 | . . . 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 1929 | . 2 ⊢ (𝜑 → (∃𝑓 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ ∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅))) |
17 | 3, 1, 5, 6, 7 | cic 17258 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓 𝑓 ∈ (𝑋(Iso‘𝐶)𝑌))) |
18 | n0 4247 | . . . . 5 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
19 | 18 | anbi1i 627 | . . . 4 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅)) |
20 | 19.41v 1958 | . . . 4 ⊢ (∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ (∃𝑓 𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅)) | |
21 | 19, 20 | bitr4i 281 | . . 3 ⊢ (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ ∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅)) |
22 | 21 | a1i 11 | . 2 ⊢ (𝜑 → (((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅) ↔ ∃𝑓(𝑓 ∈ (𝑋𝐻𝑌) ∧ (𝑌𝐻𝑋) ≠ ∅))) |
23 | 16, 17, 22 | 3bitr4d 314 | 1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ((𝑋𝐻𝑌) ≠ ∅ ∧ (𝑌𝐻𝑋) ≠ ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 Isociso 17205 ≃𝑐 ccic 17254 ThinCatcthinc 45916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-supp 7882 df-cat 17125 df-cid 17126 df-sect 17206 df-inv 17207 df-iso 17208 df-cic 17255 df-thinc 45917 |
This theorem is referenced by: postc 45977 |
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