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Theorem dmiin 5105
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 4114 . . . 4  |-  F/_ x |^|_ x  e.  A  B
21nfdm 5103 . . 3  |-  F/_ x dom  |^|_ x  e.  A  B
32ssiinf 4132 . 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 4135 . . 3  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 dmss 5061 . . 3  |-  ( |^|_ x  e.  A  B  C_  B  ->  dom  |^|_ x  e.  A  B  C_  dom  B )
64, 5syl 16 . 2  |-  ( x  e.  A  ->  dom  |^|_
x  e.  A  B  C_ 
dom  B )
73, 6mprgbir 2768 1  |-  dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1725    C_ wss 3312   |^|_ciin 4086   dom cdm 4870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-iin 4088  df-br 4205  df-dm 4880
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