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Theorem dmiin 4753
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 3812 . . . 4  |-  F/_ x |^|_ x  e.  A  B
21nfdm 4751 . . 3  |-  F/_ x dom  |^|_ x  e.  A  B
32ssiinf 3830 . 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 3833 . . 3  |-  ( x  e.  A  ->  |^|_ x  e.  A  B  C_  B
)
5 dmss 4706 . . 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 2465 1  |-  dom  |^|_ x  e.  A  B  C_  |^|_ x  e.  A  dom  B
Colors of variables: wff set class
Syntax hints:    e. wcel 1463    C_ wss 3039   |^|_ciin 3782   dom cdm 4507
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-iin 3784  df-br 3898  df-dm 4517
This theorem is referenced by: (None)
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