Theorem termchom 50100

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 50096	. . 3 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)
7		neq0 4305	. . 3 ⊢ (¬ (𝑋𝐻𝑌) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
8	6, 7	sylib 220	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
9	3	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
10	4	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
11		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌))
12	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ TermCat)
13	12	termcthind 50090	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat)
14	9, 10, 11, 2, 5, 13	thinchom 50039	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {𝑓})
15		termchom.i	. . . . 5 ⊢ 1 = (Id‘𝐶)
16	12, 2, 9, 10, 5, 11, 15	termcid 50098	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 = ( 1 ‘𝑋))
17	16	sneqd 4595	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → {𝑓} = {( 1 ‘𝑋)})
18	14, 17	eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
19	8, 18	exlimddv 1956	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1561  ∃wex 1800   ∈ wcel 2143  ∅c0 4286  {csn 4583  ‘cfv 6521  (class class class)co 7396  Basecbs 17255  Hom chom 17307  Idccid 17707  TermCatctermc 50084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-cat 17710  df-cid 17711  df-thinc 50030  df-termc 50085

This theorem is referenced by:  termchom2  50101  termcfuncval  50144