Theorem iinss2 4019

Description: An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
iinss2	⊢ (x ∈ A → ∩x ∈ A B ⊆ B)

Proof of Theorem iinss2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2863	. . . . 5 ⊢ y ∈ V
2		eliin 3975	. . . . 5 ⊢ (y ∈ V → (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B))
3	1, 2	ax-mp 5	. . . 4 ⊢ (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B)
4		rsp 2675	. . . 4 ⊢ (∀x ∈ A y ∈ B → (x ∈ A → y ∈ B))
5	3, 4	sylbi 187	. . 3 ⊢ (y ∈ ∩x ∈ A B → (x ∈ A → y ∈ B))
6	5	com12 27	. 2 ⊢ (x ∈ A → (y ∈ ∩x ∈ A B → y ∈ B))
7	6	ssrdv 3279	1 ⊢ (x ∈ A → ∩x ∈ A B ⊆ B)

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ⊆ wss 3258  ∩ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-iin 3973

This theorem is referenced by:  dmiin  4966