Theorem iinssiin 45590

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 4961	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵
3	2	nfcri 2895	. . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵
4	1, 3	nfan 1907	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵)
5		iinssiin.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
6	5	adantlr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)
7		eliinid 45572	. . . . . . 7 ⊢ ((𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
8	7	adantll 721	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐵)
9	6, 8	sseldd 3918	. . . . 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 3239	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
12		eliin 4929	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
13	12	elv 3438	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
14	11, 13	sylibr 236	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)
15	14	ssd 45543	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  Ⅎwnf 1791   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ⊆ wss 3885  ∩ ciin 4925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-v 3435  df-ss 3902  df-iin 4927

This theorem is referenced by: (None)