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Theorem dmiin 4922
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 3934 . . . 4  |-  F/_ x |^|_ x  e.  A  B
21nfdm 4920 . . 3  |-  F/_ x dom  |^|_ x  e.  A  B
32ssiinf 3951 . 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 3954 . . 3  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 dmss 4878 . . 3  |-  ( |^|_ x  e.  A  B  C_  B  ->  dom  |^|_ x  e.  A  B  C_  dom  B )
64, 5syl 15 . 2  |-  ( x  e.  A  ->  dom  |^|_
x  e.  A  B  C_ 
dom  B )
73, 6mprgbir 2613 1  |-  dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1684    C_ wss 3152   |^|_ciin 3906   dom cdm 4689
This theorem is referenced by:  domintrefc  25125  rnintintrn  25126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-iin 3908  df-br 4024  df-dm 4699
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