Theorem iinss1 4533

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 3666	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 → ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		vex 3203	. . . 4 ⊢ 𝑦 ∈ V
3		eliin 4525	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝐶)
5		eliin 4525	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
6	2, 5	ax-mp 5	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
7	1, 4, 6	3imtr4g 285	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
8	7	ssrdv 3609	1 ⊢ (𝐴 ⊆ 𝐵 → ∩ 𝑥 ∈ 𝐵 𝐶 ⊆ ∩ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ∩ ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-iin 4523

This theorem is referenced by:  polcon3N  35203  smflimsuplem5  41030  smflimsuplem7  41032