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Theorem dmiin 4681
Description: Domain of an intersection. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
dmiin  |-  dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B

Proof of Theorem dmiin
StepHypRef Expression
1 nfii1 3761 . . . 4  |-  F/_ x |^|_ x  e.  A  B
21nfdm 4679 . . 3  |-  F/_ x dom  |^|_ x  e.  A  B
32ssiinf 3779 . 2  |-  ( dom  |^|_ x  e.  A  B  C_ 
|^|_ x  e.  A  dom  B  <->  A. x  e.  A  dom  |^|_ x  e.  A  B  C_  dom  B )
4 iinss2 3782 . . 3  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 dmss 4635 . . 3  |-  ( |^|_ x  e.  A  B  C_  B  ->  dom  |^|_ x  e.  A  B  C_  dom  B )
64, 5syl 14 . 2  |-  ( x  e.  A  ->  dom  |^|_
x  e.  A  B  C_ 
dom  B )
73, 6mprgbir 2433 1  |-  dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1438    C_ wss 2999   |^|_ciin 3731   dom cdm 4438
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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-iin 3733  df-br 3846  df-dm 4448
This theorem is referenced by: (None)
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