Theorem thinchom 49438

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 49437	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐹)
13	12, 6	eqsnd 4780	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  {csn 4574  ‘cfv 6477  (class class class)co 7341  Basecbs 17112  Hom chom 17164  ThinCatcthinc 49428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-iota 6433  df-fv 6485  df-ov 7344  df-thinc 49429

This theorem is referenced by:  termchom  49499