Theorem termchommo 50322

Description: All morphisms of a terminal category are identical. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)
termcbasmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcbasmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termcid.h	⊢ 𝐻 = (Hom ‘𝐶)
termcid.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
termchommo.x	⊢ (𝜑 → 𝑍 ∈ 𝐵)
termchommo.y	⊢ (𝜑 → 𝑊 ∈ 𝐵)
termchommo.f	⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑊))

Assertion

Ref	Expression
termchommo	⊢ (𝜑 → 𝐹 = 𝐺)

Proof of Theorem termchommo

Step	Hyp	Ref	Expression
1		termcbasmo.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		termcbasmo.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		termcid.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		termchommo.f	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑊))
5		termcbas.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ TermCat)
6		termcbas.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
7		termchommo.x	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8	5, 6, 1, 7	termcbasmo 50320	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑍)
9		termchommo.y	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
10	5, 6, 2, 9	termcbasmo 50320	. . . 4 ⊢ (𝜑 → 𝑌 = 𝑊)
11	8, 10	oveq12d 7437	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑍𝐻𝑊))
12	4, 11	eleqtrrd 2868	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
13		termcid.h	. 2 ⊢ 𝐻 = (Hom ‘𝐶)
14	5	termcthind 50315	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
15	1, 2, 3, 12, 6, 13, 14	thincmo2 50263	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  Hom chom 17345  TermCatctermc 50309

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-nul 5271

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-thinc 50255  df-termc 50310

This theorem is used by:  termcarweu  50365