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Theorem dfdom2 7162
Description: Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
dfdom2  |-  ~<_  =  ( 
~<  u.  ~~  )

Proof of Theorem dfdom2
StepHypRef Expression
1 df-sdom 7141 . . 3  |-  ~<  =  (  ~<_  \  ~~  )
21uneq2i 3484 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~~  u.  (  ~<_  \  ~~  ) )
3 uncom 3477 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~<  u.  ~~  )
4 enssdom 7161 . . 3  |-  ~~  C_  ~<_
5 undif 3732 . . 3  |-  (  ~~  C_  ~<_  <-> 
(  ~~  u.  (  ~<_  \ 
~~  ) )  =  ~<_  )
64, 5mpbi 201 . 2  |-  (  ~~  u.  (  ~<_  \  ~~  )
)  =  ~<_
72, 3, 63eqtr3ri 2471 1  |-  ~<_  =  ( 
~<  u.  ~~  )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    \ cdif 3303    u. cun 3304    C_ wss 3306    ~~ cen 7135    ~<_ cdom 7136    ~< csdm 7137
This theorem is referenced by:  brdom2  7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-opab 4292  df-xp 4913  df-rel 4914  df-f1o 5490  df-en 7139  df-dom 7140  df-sdom 7141
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