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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 17622 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) |
7 | 4 | adantr 482 | . . . . . 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 2733 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
10 | 5 | adantr 482 | . . . . . 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 17623 | . . . . 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 17623 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)ℎ) = ℎ) |
15 | 12, 14 | eqeq12d 2749 | . . . 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 228 | . . 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 3204 | . 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 17677 | . 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 712 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 〈cop 4633 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 Hom chom 17204 compcco 17205 Catccat 17604 Idccid 17605 Monocmon 17671 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-cat 17608 df-cid 17609 df-mon 17673 |
This theorem is referenced by: (None) |
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