Theorem termchommo 49103

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 49101	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑍)
9		termchommo.y	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
10	5, 6, 2, 9	termcbasmo 49101	. . . 4 ⊢ (𝜑 → 𝑌 = 𝑊)
11	8, 10	oveq12d 7447	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑍𝐻𝑊))
12	4, 11	eleqtrrd 2843	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
13		termcid.h	. 2 ⊢ 𝐻 = (Hom ‘𝐶)
14	5	termcthind 49098	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
15	1, 2, 3, 12, 6, 13, 14	thincmo2 49049	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6559  (class class class)co 7429  Basecbs 17243  Hom chom 17304  TermCatctermc 49092

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-nul 5304

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-ov 7432  df-thinc 49041  df-termc 49093

This theorem is referenced by: (None)