Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  domintrefb Unicode version

Theorem domintrefb 25166
Description: The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4902. (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 3402 . . . . . . 7  |-  ( dom 
R  i^i  dom  S ) 
C_  dom  R
21sseli 3189 . . . . . 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 2628 . . . 4  |-  ( A. x  e.  dom  R  x R x  ->  A. x  e.  ( dom  R  i^i  dom 
S ) x R x )
5 inss2 3403 . . . . . . 7  |-  ( dom 
R  i^i  dom  S ) 
C_  dom  S
65sseli 3189 . . . . . 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 2628 . . . 4  |-  ( A. x  e.  dom  S  x S x  ->  A. x  e.  ( dom  R  i^i  dom 
S ) x S x )
9 r19.26 2688 . . . . 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 4086 . . . . . . . 8  |-  ( x ( R  i^i  S
) x  <->  ( x R x  /\  x S x ) )
11 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
1211, 11breldm 4899 . . . . . . . 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 2631 . . . . . 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 3183 . . . . . 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 4902 . . 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 3207 . 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 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    C_ wss 3165   class class class wbr 4039   dom cdm 4705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-dm 4715
  Copyright terms: Public domain W3C validator