Theorem termchom 49474

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 49470	. . 3 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)
7		neq0 4315	. . 3 ⊢ (¬ (𝑋𝐻𝑌) = ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
8	6, 7	sylib 218	. 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌))
9	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ∈ 𝐵)
10	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑌 ∈ 𝐵)
11		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 ∈ (𝑋𝐻𝑌))
12	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ TermCat)
13	12	termcthind 49464	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝐶 ∈ ThinCat)
14	9, 10, 11, 2, 5, 13	thinchom 49413	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {𝑓})
15		termchom.i	. . . . 5 ⊢ 1 = (Id‘𝐶)
16	12, 2, 9, 10, 5, 11, 15	termcid 49472	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑓 = ( 1 ‘𝑋))
17	16	sneqd 4601	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → {𝑓} = {( 1 ‘𝑋)})
18	14, 17	eqtrd 2764	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑋𝐻𝑌)) → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})
19	8, 18	exlimddv 1935	1 ⊢ (𝜑 → (𝑋𝐻𝑌) = {( 1 ‘𝑋)})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∅c0 4296  {csn 4589  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  Idccid 17626  TermCatctermc 49458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-cat 17629  df-cid 17630  df-thinc 49404  df-termc 49459

This theorem is referenced by:  termchom2  49475  termcfuncval  49518