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Theorem dmiin 4834
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 3882 . . . 4  |-  F/_ x |^|_ x  e.  A  B
21nfdm 4832 . . 3  |-  F/_ x dom  |^|_ x  e.  A  B
32ssiinf 3900 . 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 3903 . . 3  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 dmss 4787 . . 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 2515 1  |-  dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
Colors of variables: wff set class
Syntax hints:    e. wcel 2128    C_ wss 3102   |^|_ciin 3852   dom cdm 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-iin 3854  df-br 3968  df-dm 4598
This theorem is referenced by: (None)
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