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Mirrors > Home > MPE Home > Th. List > Mathboxes > idmon | Structured version Visualization version GIF version |
Description: An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.) |
Ref | Expression |
---|---|
idmon.b | ⊢ 𝐵 = (Base‘𝐶) |
idmon.h | ⊢ 𝐻 = (Hom ‘𝐶) |
idmon.i | ⊢ 1 = (Id‘𝐶) |
idmon.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
idmon.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
idmon.m | ⊢ 𝑀 = (Mono‘𝐶) |
Ref | Expression |
---|---|
idmon | ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idmon.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | idmon.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
3 | idmon.i | . . 3 ⊢ 1 = (Id‘𝐶) | |
4 | idmon.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | idmon.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | catidcl 17068 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) |
7 | 4 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat) |
8 | simpr1 1195 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵) | |
9 | eqid 2739 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
10 | 5 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵) |
11 | simpr2 1196 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋)) | |
12 | 1, 2, 3, 7, 8, 9, 10, 11 | catlid 17069 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)𝑔) = 𝑔) |
13 | simpr3 1197 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ℎ ∈ (𝑧𝐻𝑋)) | |
14 | 1, 2, 3, 7, 8, 9, 10, 13 | catlid 17069 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)ℎ) = ℎ) |
15 | 12, 14 | eqeq12d 2755 | . . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ((( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)𝑔) = (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)ℎ) ↔ 𝑔 = ℎ)) |
16 | 15 | biimpd 232 | . . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ((( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)𝑔) = (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)ℎ) → 𝑔 = ℎ)) |
17 | 16 | ralrimivvva 3105 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)𝑔) = (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)ℎ) → 𝑔 = ℎ)) |
18 | idmon.m | . . 3 ⊢ 𝑀 = (Mono‘𝐶) | |
19 | 1, 2, 9, 18, 4, 5, 5 | ismon2 17121 | . 2 ⊢ (𝜑 → (( 1 ‘𝑋) ∈ (𝑋𝑀𝑋) ↔ (( 1 ‘𝑋) ∈ (𝑋𝐻𝑋) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)𝑔) = (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)ℎ) → 𝑔 = ℎ)))) |
20 | 6, 17, 19 | mpbir2and 713 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 〈cop 4532 ‘cfv 6349 (class class class)co 7182 Basecbs 16598 Hom chom 16691 compcco 16692 Catccat 17050 Idccid 17051 Monocmon 17115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-1st 7726 df-2nd 7727 df-cat 17054 df-cid 17055 df-mon 17117 |
This theorem is referenced by: (None) |
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