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Theorem domintrefb 25063
Description: The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4886. (Contributed by FL, 30-Dec-2011.)
Assertion
Ref Expression
domintrefb  |-  ( ( A. x  e.  dom  R  x R x  /\  A. x  e.  dom  S  x S x )  ->  dom  ( R  i^i  S
)  =  ( dom 
R  i^i  dom  S ) )
Distinct variable groups:    x, R    x, S

Proof of Theorem domintrefb
StepHypRef Expression
1 inss1 3389 . . . . . . 7  |-  ( dom 
R  i^i  dom  S ) 
C_  dom  R
21sseli 3176 . . . . . 6  |-  ( x  e.  ( dom  R  i^i  dom  S )  ->  x  e.  dom  R )
32imim1i 54 . . . . 5  |-  ( ( x  e.  dom  R  ->  x R x )  ->  ( x  e.  ( dom  R  i^i  dom 
S )  ->  x R x ) )
43ralimi2 2615 . . . 4  |-  ( A. x  e.  dom  R  x R x  ->  A. x  e.  ( dom  R  i^i  dom 
S ) x R x )
5 inss2 3390 . . . . . . 7  |-  ( dom 
R  i^i  dom  S ) 
C_  dom  S
65sseli 3176 . . . . . 6  |-  ( x  e.  ( dom  R  i^i  dom  S )  ->  x  e.  dom  S )
76imim1i 54 . . . . 5  |-  ( ( x  e.  dom  S  ->  x S x )  ->  ( x  e.  ( dom  R  i^i  dom 
S )  ->  x S x ) )
87ralimi2 2615 . . . 4  |-  ( A. x  e.  dom  S  x S x  ->  A. x  e.  ( dom  R  i^i  dom 
S ) x S x )
9 r19.26 2675 . . . . 5  |-  ( A. x  e.  ( dom  R  i^i  dom  S )
( x R x  /\  x S x )  <->  ( A. x  e.  ( dom  R  i^i  dom 
S ) x R x  /\  A. x  e.  ( dom  R  i^i  dom 
S ) x S x ) )
10 brin 4070 . . . . . . . 8  |-  ( x ( R  i^i  S
) x  <->  ( x R x  /\  x S x ) )
11 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
1211, 11breldm 4883 . . . . . . . 8  |-  ( x ( R  i^i  S
) x  ->  x  e.  dom  ( R  i^i  S ) )
1310, 12sylbir 204 . . . . . . 7  |-  ( ( x R x  /\  x S x )  ->  x  e.  dom  ( R  i^i  S ) )
1413ralimi 2618 . . . . . 6  |-  ( A. x  e.  ( dom  R  i^i  dom  S )
( x R x  /\  x S x )  ->  A. x  e.  ( dom  R  i^i  dom 
S ) x  e. 
dom  ( R  i^i  S ) )
15 dfss3 3170 . . . . . 6  |-  ( ( dom  R  i^i  dom  S )  C_  dom  ( R  i^i  S )  <->  A. x  e.  ( dom  R  i^i  dom 
S ) x  e. 
dom  ( R  i^i  S ) )
1614, 15sylibr 203 . . . . 5  |-  ( A. x  e.  ( dom  R  i^i  dom  S )
( x R x  /\  x S x )  ->  ( dom  R  i^i  dom  S )  C_ 
dom  ( R  i^i  S ) )
179, 16sylbir 204 . . . 4  |-  ( ( A. x  e.  ( dom  R  i^i  dom  S ) x R x  /\  A. x  e.  ( dom  R  i^i  dom 
S ) x S x )  ->  ( dom  R  i^i  dom  S
)  C_  dom  ( R  i^i  S ) )
184, 8, 17syl2an 463 . . 3  |-  ( ( A. x  e.  dom  R  x R x  /\  A. x  e.  dom  S  x S x )  -> 
( dom  R  i^i  dom 
S )  C_  dom  ( R  i^i  S ) )
19 dmin 4886 . . 3  |-  dom  ( R  i^i  S )  C_  ( dom  R  i^i  dom  S )
2018, 19jctil 523 . 2  |-  ( ( A. x  e.  dom  R  x R x  /\  A. x  e.  dom  S  x S x )  -> 
( dom  ( R  i^i  S )  C_  ( dom  R  i^i  dom  S
)  /\  ( dom  R  i^i  dom  S )  C_ 
dom  ( R  i^i  S ) ) )
21 eqss 3194 . 2  |-  ( dom  ( R  i^i  S
)  =  ( dom 
R  i^i  dom  S )  <-> 
( dom  ( R  i^i  S )  C_  ( dom  R  i^i  dom  S
)  /\  ( dom  R  i^i  dom  S )  C_ 
dom  ( R  i^i  S ) ) )
2220, 21sylibr 203 1  |-  ( ( A. x  e.  dom  R  x R x  /\  A. x  e.  dom  S  x S x )  ->  dom  ( R  i^i  S
)  =  ( dom 
R  i^i  dom  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   class class class wbr 4023   dom cdm 4689
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-br 4024  df-dm 4699
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