Theorem iinssiin 43818

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	. . . . 5 ⊢ Ⅎ𝑥𝜑
2		nfii1 5033	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
3	2	nfcri 2891	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵
4	1, 3	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
5		iinssiin.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
6	5	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
7		eliinid 43800	. . . . . . 7 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
8	7	adantll 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
9	6, 8	sseldd 3984	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐶)
10	9	ex 414	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐶))
11	4, 10	ralrimi 3255	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
12		eliin 5003	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
13	12	elv 3481	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
14	11, 13	sylibr 233	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
15	14	ssd 43769	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  Ⅎwnf 1786   ∈ wcel 2107  ∀wral 3062  Vcvv 3475   ⊆ wss 3949  ∩ ciin 4999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-v 3477  df-in 3956  df-ss 3966  df-iin 5001

This theorem is referenced by: (None)