Theorem termchom 50294

Description: The hom-set of a terminal category is a singleton of the identity morphism. (Contributed by Zhi Wang, 20-Oct-2025.)

Hypotheses

Ref	Expression
termchom.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termchom.b	⊢ 𝐵 = (Base‘𝐶)
termchom.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termchom.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termchom.h	⊢ 𝐻 = (Hom ‘𝐶)
termchom.i	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
termchom	⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})

Proof of Theorem termchom

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termchom.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termchom.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3		termchom.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		termchom.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		termchom.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
6	1, 2, 3, 4, 5	termchomn0 50290	. . 3 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)
7		neq0 4305	. . 3 ⊢ (¬ (𝑋𝐻𝑌) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
8	6, 7	sylib 221	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
9	3	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
10	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
11		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌))
12	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ TermCat)
13	12	termcthind 50284	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat)
14	9, 10, 11, 2, 5, 13	thinchom 50233	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {𝑓})
15		termchom.i	. . . . 5 ⊢ 1 = (Id‘𝐶)
16	12, 2, 9, 10, 5, 11, 15	termcid 50292	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 = ( 1 ‘𝑋))
17	16	sneqd 4600	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → {𝑓} = {( 1 ‘𝑋)})
18	14, 17	eqtrd 2797	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
19	8, 18	exlimddv 1964	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  ∅c0 4285  {csn 4588  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  Hom chom 17327  Idccid 17727  TermCatctermc 50278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-cat 17730  df-cid 17731  df-thinc 50224  df-termc 50279

This theorem is used by:  termchom2  50295  termcfuncval  50338