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Theorem dfiin3g 4691
Description: Alternate definition of indexed intersection when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfiin3g  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )

Proof of Theorem dfiin3g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfiin2g 3763 . 2  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| { y  |  E. x  e.  A  y  =  B } )
2 eqid 2088 . . . 4  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
32rnmpt 4683 . . 3  |-  ran  (
x  e.  A  |->  B )  =  { y  |  E. x  e.  A  y  =  B }
43inteqi 3692 . 2  |-  |^| ran  ( x  e.  A  |->  B )  =  |^| { y  |  E. x  e.  A  y  =  B }
51, 4syl6eqr 2138 1  |-  ( A. x  e.  A  B  e.  C  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   |^|cint 3688   |^|_ciin 3731    |-> cmpt 3899   ran crn 4439
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-int 3689  df-iin 3733  df-br 3846  df-opab 3900  df-mpt 3901  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  dfiin3  4693  riinint  4694
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