Theorem viin 4025

Description: Indexed intersection with a universal index class. When A doesn't depend on x, this evaluates to A by 19.3 1785 and abid2 2470. When A = x, this evaluates to ∅ by intiin 4020 and intv in set.mm. (Contributed by NM, 11-Sep-2008.)

Assertion

Ref	Expression
viin	⊢ ∩x ∈ V A = {y ∣ ∀x y ∈ A}

Distinct variable groups:   x,y   y,A

Allowed substitution hint:   A(x)

Proof of Theorem viin

Step	Hyp	Ref	Expression
1		df-iin 3972	. 2 ⊢ ∩x ∈ V A = {y ∣ ∀x ∈ V y ∈ A}
2		ralv 2872	. . 3 ⊢ (∀x ∈ V y ∈ A ↔ ∀x y ∈ A)
3	2	abbii 2465	. 2 ⊢ {y ∣ ∀x ∈ V y ∈ A} = {y ∣ ∀x y ∈ A}
4	1, 3	eqtri 2373	1 ⊢ ∩x ∈ V A = {y ∣ ∀x y ∈ A}

Syntax hints:  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  Vcvv 2859  ∩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-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ral 2619  df-v 2861  df-iin 3972

This theorem is referenced by: (None)