Theorem idmon 49009

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 17643	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
7	4	adantr 480	. . . . . 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 2729	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	5	adantr 480	. . . . . 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 17644	. . . . 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 17644	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)ℎ) = ℎ)
15	12, 14	eqeq12d 2745	. . . 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 229	. . 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 3183	. 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 17696	. 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))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟨cop 4595  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  compcco 17232  Catccat 17625  Idccid 17626  Monocmon 17690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-cat 17629  df-cid 17630  df-mon 17692

This theorem is referenced by: (None)