Theorem thinchom 49280

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 49279	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐹)
13	12, 6	eqsnd 4811	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4606  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  Hom chom 17287  ThinCatcthinc 49270

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-nul 5281

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413  df-thinc 49271

This theorem is referenced by:  termchom  49340