Theorem iinssd 45490

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

Hypotheses

Ref	Expression
iinssd.1	⊢ (𝜑 → 𝑋 ∈ 𝐴)
iinssd.2	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷)
iinssd.3	⊢ (𝜑 → 𝐷 ⊆ 𝐶)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑋

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem iinssd

Step	Hyp	Ref	Expression
1		iinssd.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
2		iinssd.3	. . 3 ⊢ (𝜑 → 𝐷 ⊆ 𝐶)
3		iinssd.2	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐷)
4	3	sseq1d 3967	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ⊆ 𝐶 ↔ 𝐷 ⊆ 𝐶))
5	4	rspcev 3578	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶) → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
6	1, 2, 5	syl2anc 585	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
7		iinss 5014	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)
8	6, 7	syl 17	1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   ⊆ wss 3903  ∩ ciin 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-v 3444  df-ss 3920  df-iin 4951

This theorem is referenced by:  smfsuplem3  47171  smflimsuplem1  47178