Theorem termcbas2 49493

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 49491	. 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 2831	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝑋 ∈ {𝑥})
8		elsni 4591	. . . . 5 ⊢ (𝑋 ∈ {𝑥} → 𝑋 = 𝑥)
9	8	sneqd 4586	. . . 4 ⊢ (𝑋 ∈ {𝑥} → {𝑋} = {𝑥})
10	7, 9	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → {𝑋} = {𝑥})
11	4, 10	eqtr4d 2768	. 2 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑋})
12	3, 11	exlimddv 1936	1 ⊢ (𝜑 → 𝐵 = {𝑋})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  {csn 4574  ‘cfv 6477  Basecbs 17112  TermCatctermc 49483

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 2112  ax-9 2120  ax-ext 2702

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-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-iota 6433  df-fv 6485  df-termc 49484

This theorem is referenced by:  termcfuncval  49543  diag1f1olem  49544  termcnatval  49546  diag2f1olem  49547