Theorem iinssiin 40121

Description: Subset implication for an indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
iinssiin.1	⊢ Ⅎ𝑥𝜑
iinssiin.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
iinssiin	⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Proof of Theorem iinssiin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iinssiin.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
2		nfcv 2969	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
3		nfii1 4773	. . . . . . 7 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
4	2, 3	nfel 2982	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵
5	1, 4	nfan 2002	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
6		iinssiin.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
7	6	adantlr 706	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
8		eliinid 40104	. . . . . . . 8 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
9	8	adantll 705	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
10	7, 9	sseldd 3828	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐶)
11	10	ex 403	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐶))
12	5, 11	ralrimi 3166	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
13		vex 3417	. . . . 5 ⊢ 𝑦 ∈ V
14		eliin 4747	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
15	13, 14	ax-mp 5	. . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
16	12, 15	sylibr 226	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
17	16	ralrimiva 3175	. 2 ⊢ (𝜑 → ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
18		dfss3 3816	. 2 ⊢ (∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
19	17, 18	sylibr 226	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  Ⅎwnf 1882   ∈ wcel 2164  ∀wral 3117  Vcvv 3414   ⊆ wss 3798  ∩ ciin 4743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-v 3416  df-in 3805  df-ss 3812  df-iin 4745

This theorem is referenced by: (None)