Theorem idmon 46185

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 17308	. 2 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋))
7	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ Cat)
8		simpr1 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
9		eqid 2738	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
10	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
11		simpr2 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
12	1, 2, 3, 7, 8, 9, 10, 11	catlid 17309	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → (( 1 ‘𝑋)(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑋)𝑔) = 𝑔)
13		simpr3 1194	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ℎ ∈ (𝑧𝐻𝑋))
14	1, 2, 3, 7, 8, 9, 10, 13	catlid 17309	. . . . 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 3115	. 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 17363	. 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 709	1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝑀𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Idccid 17291  Monocmon 17357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-cat 17294  df-cid 17295  df-mon 17359

This theorem is referenced by: (None)