Theorem iinss1 4974

Description: Subclass theorem for indexed intersection. (Contributed by NM, 24-Jan-2012.)

Assertion

Ref	Expression
iinss1	⊢ (𝐴 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinss1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssralv 4018	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		eliin 4963	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
3	2	elv 3455	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
4		eliin 4963	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	elv 3455	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6	1, 3, 5	3imtr4g 296	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
7	6	ssrdv 3955	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ∩ ciin 4959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-ss 3934  df-iin 4961

This theorem is referenced by:  polcon3N  39918  smflimsuplem5  46829  smflimsuplem7  46831