Theorem iinssiin 45087

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 5003	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
3	2	nfcri 2889	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵
4	1, 3	nfan 1898	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
5		iinssiin.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
6	5	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
7		eliinid 45069	. . . . . . 7 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
8	7	adantll 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
9	6, 8	sseldd 3957	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐶)
10	9	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐶))
11	4, 10	ralrimi 3238	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
12		eliin 4970	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
13	12	elv 3462	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
14	11, 13	sylibr 234	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
15	14	ssd 45038	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  Ⅎwnf 1782   ∈ wcel 2107  ∀wral 3050  Vcvv 3457   ⊆ wss 3924  ∩ ciin 4966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-v 3459  df-ss 3941  df-iin 4968

This theorem is referenced by: (None)