Theorem dfiin2 3921

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 3919	. 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 2534	1 |- |^|_ x e. A B = |^| { y | E. x e. A y = B }

Syntax hints:   = wceq 1353   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2737   |^|cint 3844   |^|_ciin 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-int 3845  df-iin 3889

This theorem is referenced by: (None)