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Theorem dfiin3 4908
Description: Alternate definition of indexed intersection when  B is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfiun3.1  |-  B  e. 
_V
Assertion
Ref Expression
dfiin3  |-  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B )

Proof of Theorem dfiin3
StepHypRef Expression
1 dfiin3g 4906 . 2  |-  ( A. x  e.  A  B  e.  _V  ->  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B ) )
2 dfiun3.1 . . 3  |-  B  e. 
_V
32a1i 9 . 2  |-  ( x  e.  A  ->  B  e.  _V )
41, 3mprg 2547 1  |-  |^|_ x  e.  A  B  =  |^| ran  ( x  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   _Vcvv 2752   |^|cint 3862   |^|_ciin 3905    |-> cmpt 4082   ran crn 4648
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-int 3863  df-iin 3907  df-br 4022  df-opab 4083  df-mpt 4084  df-cnv 4655  df-dm 4657  df-rn 4658
This theorem is referenced by: (None)
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