Theorem idmon 50339

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 17885	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
7	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
8		simpr1 1213	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
9		eqid 2763	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
11		simpr2 1214	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
12	1, 2, 3, 7, 8, 9, 10, 11	catlid 17886	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
13		simpr3 1215	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ℎ ∈ (𝑧𝐻𝑋))
14	1, 2, 3, 7, 8, 9, 10, 13	catlid 17886	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)ℎ) = ℎ)
15	12, 14	eqeq12d 2779	. . . 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 3212	. 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 17938	. 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 726	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ⟨cop 4838  ‘cfv 6558  (class class class)co 7443  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868  Monocmon 17932

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-cat 17871  df-cid 17872  df-mon 17934

This theorem is used by: (None)