Theorem termchom 50518

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 50514	. . 3 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)
7		neq0 4298	. . 3 ⊢ (¬ (𝑋𝐻𝑌) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
8	6, 7	sylib 221	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
9	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
10	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
11		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌))
12	1	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ TermCat)
13	12	termcthind 50508	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat)
14	9, 10, 11, 2, 5, 13	thinchom 50457	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {𝑓})
15		termchom.i	. . . . 5 ⊢ 1 = (Id‘𝐶)
16	12, 2, 9, 10, 5, 11, 15	termcid 50516	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 = ( 1 ‘𝑋))
17	16	sneqd 4595	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → {𝑓} = {( 1 ‘𝑋)})
18	14, 17	eqtrd 2795	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
19	8, 18	exlimddv 1968	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∅c0 4278  {csn 4583  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  Hom chom 17400  Idccid 17800  TermCatctermc 50502

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-cat 17803  df-cid 17804  df-thinc 50448  df-termc 50503

This theorem is used by:  termchom2  50519  termcfuncval  50562