MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiin2 Structured version   Unicode version

Theorem dfiin2 4270
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
StepHypRef Expression
1 dfiin2g 4268 . 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
32a1i 11 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2722 1  |-  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1872   {cab 2408   E.wrex 2709   _Vcvv 3016   |^|cint 4191   |^|_ciin 4236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ral 2713  df-rex 2714  df-v 3018  df-int 4192  df-iin 4238
This theorem is referenced by:  fniinfv  5877  scott0  8302  cfval2  8634  cflim3  8636  cflim2  8637  cfss  8639  hauscmplem  20356  ptbasfi  20531  dihglblem5  34772  dihglb2  34816  intima0  36146
  Copyright terms: Public domain W3C validator