Theorem iinss1 3981

Description: Subclass theorem for indexed union. (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 3330	. . 3 ⊢ (A ⊆ B → (∀x ∈ B y ∈ C → ∀x ∈ A y ∈ C))
2		vex 2862	. . . 4 ⊢ y ∈ V
3		eliin 3974	. . . 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 3974	. . . 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 261	. 2 ⊢ (A ⊆ B → (y ∈ ∩x ∈ B C → y ∈ ∩x ∈ A C))
8	7	ssrdv 3278	1 ⊢ (A ⊆ B → ∩x ∈ B C ⊆ ∩x ∈ A C)

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ⊆ wss 3257  ∩ciin 3970

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 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iin 3972

This theorem is referenced by: (None)