Theorem dfiin2 4003

Description: Alternate definition of indexed intersection when B is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Hypothesis

Ref	Expression
dfiun2.1	⊢ B ∈ V

Assertion

Ref	Expression
dfiin2	⊢ ∩x ∈ A B = ∩{y ∣ ∃x ∈ A y = B}

Distinct variable groups:   x,y   y,A   y,B

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem dfiin2

Step	Hyp	Ref	Expression
1		dfiin2g 4001	. 2 ⊢ (∀x ∈ A B ∈ V → ∩x ∈ A B = ∩{y ∣ ∃x ∈ A y = B})
2		dfiun2.1	. . 3 ⊢ B ∈ V
3	2	a1i 10	. 2 ⊢ (x ∈ A → B ∈ V)
4	1, 3	mprg 2684	1 ⊢ ∩x ∈ A B = ∩{y ∣ ∃x ∈ A y = B}

Syntax hints:   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860  ∩cint 3927  ∩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-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-rex 2621  df-v 2862  df-int 3928  df-iin 3973

This theorem is referenced by:  fniinfv  5373