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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 17391 | . 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) |
7 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat) |
8 | simpr1 1193 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵) | |
9 | eqid 2738 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
10 | 5 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵) |
11 | simpr2 1194 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋)) | |
12 | 1, 2, 3, 7, 8, 9, 10, 11 | catlid 17392 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)𝑔) = 𝑔) |
13 | simpr3 1195 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ℎ ∈ (𝑧𝐻𝑋)) | |
14 | 1, 2, 3, 7, 8, 9, 10, 13 | catlid 17392 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(〈𝑧, 𝑋〉(comp‘𝐶)𝑋)ℎ) = ℎ) |
15 | 12, 14 | eqeq12d 2754 | . . . 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 3127 | . 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 17446 | . 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 710 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 〈cop 4567 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Hom chom 16973 compcco 16974 Catccat 17373 Idccid 17374 Monocmon 17440 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-cat 17377 df-cid 17378 df-mon 17442 |
This theorem is referenced by: (None) |
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