Theorem iineqconst2 49661

Description: Indexed intersection of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)

Assertion

Ref	Expression
iineqconst2	⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iineqconst2

Step	Hyp	Ref	Expression
1		r19.2z 4462	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 = 𝐶)
2		eqimss 3996	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
3	2	reximi 3105	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iinss 5023	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
5	1, 3, 4	3syl 19	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		eqimss2 3997	. . . . 5 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
7	6	ralimi 3104	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
8	7	adantl 487	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
9		ssiin 5022	. . 3 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
10	8, 9	sylibr 237	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
11	5, 10	eqssd 3955	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2960  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  ∅c0 4286  ∩ ciin 4959

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-v 3459  df-dif 3909  df-ss 3923  df-nul 4287  df-iin 4961

This theorem is used by: (None)