Theorem idmon 49517

Description: An identity arrow, or an identity morphism, is a monomorphism. (Contributed by Zhi Wang, 21-Sep-2024.)

Hypotheses

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)

Assertion

Ref	Expression
idmon	⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))

Proof of Theorem idmon

Dummy variables 𝑔 ℎ 𝑧 are mutually distinct and distinct from all other variables.

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 17646	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
7	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
8		simpr1 1201	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
9		eqid 2740	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
11		simpr2 1202	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
12	1, 2, 3, 7, 8, 9, 10, 11	catlid 17647	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
13		simpr3 1203	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ℎ ∈ (𝑧𝐻𝑋))
14	1, 2, 3, 7, 8, 9, 10, 13	catlid 17647	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)ℎ) = ℎ)
15	12, 14	eqeq12d 2756	. . . 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 230	. . 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 3186	. 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 17699	. 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 719	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ⟨cop 4568  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  Hom chom 17229  compcco 17230  Catccat 17628  Idccid 17629  Monocmon 17693

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-cat 17632  df-cid 17633  df-mon 17695

This theorem is referenced by: (None)