Theorem thinchom 50578

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 486	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
3		thinchom.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
5		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 ∈ (𝑋𝐻𝑌))
6		thinchom.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
7	6	adantr 486	. . 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 486	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat)
12	2, 4, 5, 7, 8, 9, 11	thincmo2 50577	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐹)
13	12, 6	eqsnd 4790	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {𝐹})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {csn 4583  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  Hom chom 17439  ThinCatcthinc 50568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-thinc 50569

This theorem is used by:  termchom  50639