Theorem dfiin2 3713

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 e. _V

Assertion

Ref	Expression
dfiin2	|- |^|_ x e. A B = |^| { y | E. x e. 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 3711	. 2 |- ( A. x e. A B e. _V -> |^|_ x e. A B = |^| { y | E. x e. A y = B } )
2		dfiun2.1	. . 3 |- B e. _V
3	2	a1i 9	. 2 |- ( x e. A -> B e. _V )
4	1, 3	mprg 2420	1 |- |^|_ x e. A B = |^| { y | E. x e. A y = B }

Syntax hints:   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   _Vcvv 2601   |^|cint 3636   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-int 3637  df-iin 3681

This theorem is referenced by: (None)