Theorem thinchom 49986

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 483	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
3		thinchom.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4	3	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
5		simpr 487	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 ∈ (𝑋𝐻𝑌))
6		thinchom.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
7	6	adantr 483	. . 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 483	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat)
12	2, 4, 5, 7, 8, 9, 11	thincmo2 49985	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐹)
13	12, 6	eqsnd 4778	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1550   ∈ wcel 2132  {csn 4572  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  Hom chom 17269  ThinCatcthinc 49976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-nul 5246

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-iota 6462  df-fv 6514  df-ov 7384  df-thinc 49977

This theorem is referenced by:  termchom  50047