Theorem termchommo 49960

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 49958	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑍)
9		termchommo.y	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
10	5, 6, 2, 9	termcbasmo 49958	. . . 4 ⊢ (𝜑 → 𝑌 = 𝑊)
11	8, 10	oveq12d 7385	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑍𝐻𝑊))
12	4, 11	eleqtrrd 2839	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
13		termcid.h	. 2 ⊢ 𝐻 = (Hom ‘𝐶)
14	5	termcthind 49953	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
15	1, 2, 3, 12, 6, 13, 14	thincmo2 49901	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  TermCatctermc 49947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-thinc 49893  df-termc 49948

This theorem is referenced by:  termcarweu  50003