Theorem thincmon 46203

Description: In a thin category, all morphisms are monomorphisms. The converse does not hold. See grptcmon 46263. (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 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑧 ∈ 𝐵)
2		thincid.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑋 ∈ 𝐵)
4		simpr2 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 ∈ (𝑧𝐻𝑋))
5		simpr3 1194	. . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝐶 ∈ ThinCat)
10	1, 3, 4, 5, 6, 7, 9	thincmo2 46197	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑧𝐻𝑋) ∧ ℎ ∈ (𝑧𝐻𝑋))) → 𝑔 = ℎ)
11	10	a1d 25	. . . 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 3115	. . 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 2738	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
14		thincmon.m	. . . 4 ⊢ 𝑀 = (Mono‘𝐶)
15	8	thinccd 46194	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
16		thincmon.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
17	6, 7, 13, 14, 15, 2, 16	ismon2 17363	. . 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 704	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝑀𝑌) ↔ 𝑓 ∈ (𝑋𝐻𝑌)))
19	18	eqrdv 2736	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑋𝐻𝑌))

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  Monocmon 17357  ThinCatcthinc 46188

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-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-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-cat 17294  df-mon 17359  df-thinc 46189

This theorem is referenced by: (None)