Theorem termchommo 49517

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 49515	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑍)
9		termchommo.y	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
10	5, 6, 2, 9	termcbasmo 49515	. . . 4 ⊢ (𝜑 → 𝑌 = 𝑊)
11	8, 10	oveq12d 7359	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑍𝐻𝑊))
12	4, 11	eleqtrrd 2834	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
13		termcid.h	. 2 ⊢ 𝐻 = (Hom ‘𝐶)
14	5	termcthind 49510	. 2 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
15	1, 2, 3, 12, 6, 13, 14	thincmo2 49458	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  Hom chom 17167  TermCatctermc 49504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-nul 5239

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-iota 6432  df-fv 6484  df-ov 7344  df-thinc 49450  df-termc 49505

This theorem is referenced by:  termcarweu  49560