Theorem iineqconst2 49622

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 4460	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 = 𝐶)
2		eqimss 3995	. . . 4 ⊢ (𝐵 = 𝐶 → 𝐵 ⊆ 𝐶)
3	2	reximi 3103	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
4		iinss 5021	. . 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 3996	. . . . 5 ⊢ (𝐵 = 𝐶 → 𝐶 ⊆ 𝐵)
7	6	ralimi 3102	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
8	7	adantl 486	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∀𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵)
9		ssiin 5020	. . 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 3954	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ≠ wne 2958  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  ∅c0 4286  ∩ ciin 4957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-v 3457  df-dif 3908  df-ss 3922  df-nul 4287  df-iin 4959

This theorem is referenced by: (None)