Theorem thincmon 50091

Description: In a thin category, all morphisms are monomorphisms. Example 7.33(9) of [Adamek] p. 110. The converse does not hold. See grptcmon 50251. (Contributed by Zhi Wang, 24-Sep-2024.)

Hypotheses

Ref	Expression
thincid.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)
thincid.b	⊢ 𝐵 = (Base‘𝐶)
thincid.h	⊢ 𝐻 = (Hom ‘𝐶)
thincid.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincmon.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thincmon.m	⊢ 𝑀 = (Mono‘𝐶)

Assertion

Ref	Expression
thincmon	⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

Proof of Theorem thincmon

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

Step	Hyp	Ref	Expression
1		simpr1 1211	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
2		thincid.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
4		simpr2 1212	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
5		simpr3 1213	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ℎ ∈ (𝑧𝐻𝑋))
6		thincid.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
7		thincid.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
8		thincid.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
9	8	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ ThinCat)
10	1, 3, 4, 5, 6, 7, 9	thincmo2 50084	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 = ℎ)
11	10	a1d 26	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → ((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
12	11	ralrimivvva 3217	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))
13		eqid 2769	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
14		thincmon.m	. . . 4 ⊢ 𝑀 = (Mono‘𝐶)
15	8	thinccd 50081	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
16		thincmon.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	6, 7, 13, 14, 15, 2, 16	ismon2 17787	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)𝑔) = (𝑓(⟨𝑧, 𝑋⟩(comp‘𝐶)𝑌)ℎ) → 𝑔 = ℎ))))
18	12, 17	mpbiran2d 720	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ 𝑓 ∈ (𝑋𝐻𝑌)))
19	18	eqrdv 2767	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ⟨cop 4597  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  Hom chom 17317  compcco 17318  Monocmon 17781  ThinCatcthinc 50075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-cat 17720  df-mon 17783  df-thinc 50076

This theorem is referenced by: (None)