Theorem iinconst 3978

Description: Indexed intersection of a constant class, i.e. where B does not depend on x. (Contributed by Mario Carneiro, 6-Feb-2015.)

Assertion

Ref	Expression
iinconst	⊢ (A ≠ ∅ → ∩x ∈ A B = B)

Distinct variable groups:   x,A   x,B

Proof of Theorem iinconst

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.3rzv 3643	. . 3 ⊢ (A ≠ ∅ → (y ∈ B ↔ ∀x ∈ A y ∈ B))
2		vex 2862	. . . 4 ⊢ y ∈ V
3		eliin 3974	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B))
4	2, 3	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ A B ↔ ∀x ∈ A y ∈ B)
5	1, 4	syl6rbbr 255	. 2 ⊢ (A ≠ ∅ → (y ∈ ∩x ∈ A B ↔ y ∈ B))
6	5	eqrdv 2351	1 ⊢ (A ≠ ∅ → ∩x ∈ A B = B)

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  Vcvv 2859  ∅c0 3550  ∩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-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iin 3972

This theorem is referenced by: (None)