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Theorem dfdom2 7553
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 7531 . . 3  |-  ~<  =  (  ~<_  \  ~~  )
21uneq2i 3660 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~~  u.  (  ~<_  \  ~~  ) )
3 uncom 3653 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~<  u.  ~~  )
4 enssdom 7552 . . 3  |-  ~~  C_  ~<_
5 undif 3913 . . 3  |-  (  ~~  C_  ~<_  <-> 
(  ~~  u.  (  ~<_  \ 
~~  ) )  =  ~<_  )
64, 5mpbi 208 . 2  |-  (  ~~  u.  (  ~<_  \  ~~  )
)  =  ~<_
72, 3, 63eqtr3ri 2505 1  |-  ~<_  =  ( 
~<  u.  ~~  )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    \ cdif 3478    u. cun 3479    C_ wss 3481    ~~ cen 7525    ~<_ cdom 7526    ~< csdm 7527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512  df-xp 5011  df-rel 5012  df-f1o 5601  df-en 7529  df-dom 7530  df-sdom 7531
This theorem is referenced by:  brdom2  7557
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