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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincmon | Structured version Visualization version GIF version |
Description: In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 45991. (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
thincid.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
thincid.b | ⊢ 𝐵 = (Base‘𝐶) |
thincid.h | ⊢ 𝐻 = (Hom ‘𝐶) |
thincid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
thincmon.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
thincmon.m | ⊢ 𝑀 = (Mono‘𝐶) |
Ref | Expression |
---|---|
thincmon | ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr1 1196 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵) | |
2 | thincid.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵) |
4 | simpr2 1197 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋)) | |
5 | simpr3 1198 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ℎ ∈ (𝑧𝐻𝑋)) | |
6 | thincid.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
7 | thincid.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
8 | thincid.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
9 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ ThinCat) |
10 | 1, 3, 4, 5, 6, 7, 9 | thincmo2 45925 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 = ℎ) |
11 | 10 | a1d 25 | . . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ((𝑓(〈𝑧, 𝑋〉(comp‘𝐶)𝑌)𝑔) = (𝑓(〈𝑧, 𝑋〉(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ)) |
12 | 11 | ralrimivvva 3103 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝑓(〈𝑧, 𝑋〉(comp‘𝐶)𝑌)𝑔) = (𝑓(〈𝑧, 𝑋〉(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ)) |
13 | eqid 2736 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
14 | thincmon.m | . . . 4 ⊢ 𝑀 = (Mono‘𝐶) | |
15 | 8 | thinccd 45922 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
16 | thincmon.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | 6, 7, 13, 14, 15, 2, 16 | ismon2 17193 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝑓(〈𝑧, 𝑋〉(comp‘𝐶)𝑌)𝑔) = (𝑓(〈𝑧, 𝑋〉(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ)))) |
18 | 12, 17 | mpbiran2d 708 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ 𝑓 ∈ (𝑋𝐻𝑌))) |
19 | 18 | eqrdv 2734 | 1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 〈cop 4533 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 compcco 16761 Monocmon 17187 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-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-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-cat 17125 df-mon 17189 df-thinc 45917 |
This theorem is referenced by: (None) |
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