Theorem thinchom 49176

Description: A non-empty hom-set of a thin category is given by its element. (Contributed by Zhi Wang, 20-Oct-2025.)

Hypotheses

Ref	Expression
thinchom.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thinchom.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thinchom.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
thinchom.b	⊢ 𝐵 = (Base‘𝐶)
thinchom.h	⊢ 𝐻 = (Hom ‘𝐶)
thinchom.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Assertion

Ref	Expression
thinchom	⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})

Proof of Theorem thinchom

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		thinchom.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
3		thinchom.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
5		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 ∈ (𝑋𝐻𝑌))
6		thinchom.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋𝐻𝑌))
8		thinchom.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
9		thinchom.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
10		thinchom.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat)
12	2, 4, 5, 7, 8, 9, 11	thincmo2 49175	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐹)
13	12, 6	eqsnd 4804	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {csn 4599  ‘cfv 6528  (class class class)co 7400  Basecbs 17215  Hom chom 17269  ThinCatcthinc 49166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-nul 5274

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-iota 6481  df-fv 6536  df-ov 7403  df-thinc 49167

This theorem is referenced by:  termchom  49234