Theorem termcbas2 49945

Description: The base of a terminal category is given by its object. (Contributed by Zhi Wang, 20-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)
termcbasmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
termcbas2	⊢ (𝜑 → 𝐵 = {𝑋})

Proof of Theorem termcbas2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	termcbas 49943	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑥})
5		termcbasmo.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝑋 ∈ 𝐵)
7	6, 4	eleqtrd 2837	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝑋 ∈ {𝑥})
8		elsni 4574	. . . . 5 ⊢ (𝑋 ∈ {𝑥} → 𝑋 = 𝑥)
9	8	sneqd 4569	. . . 4 ⊢ (𝑋 ∈ {𝑥} → {𝑋} = {𝑥})
10	7, 9	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → {𝑋} = {𝑥})
11	4, 10	eqtr4d 2773	. 2 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑋})
12	3, 11	exlimddv 1937	1 ⊢ (𝜑 → 𝐵 = {𝑋})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4557  ‘cfv 6487  Basecbs 17168  TermCatctermc 49935

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-ext 2707

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-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6443  df-fv 6495  df-termc 49936

This theorem is referenced by:  termcfuncval  49995  diag1f1olem  49996  termcnatval  49998  diag2f1olem  49999