Theorem iineqconst2 49445

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 4453	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 = 𝐶)
2		eqimss 3994	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
3	2	reximi 3100	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iinss 5014	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
5	1, 3, 4	3syl 18	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6		eqimss2 3995	. . . . 5 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
7	6	ralimi 3099	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
8	7	adantl 485	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
9		ssiin 5013	. . 3 ⊢ (𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
10	8, 9	sylibr 236	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
11	5, 10	eqssd 3953	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ≠ wne 2957  ∀wral 3076  ∃wrex 3086   ⊆ wss 3904  ∅c0 4285  ∩ ciin 4950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286  df-iin 4952

This theorem is referenced by: (None)