MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdom2 Unicode version

Theorem dfdom2 7096
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 7075 . . 3  |-  ~<  =  (  ~<_  \  ~~  )
21uneq2i 3462 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~~  u.  (  ~<_  \  ~~  ) )
3 uncom 3455 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~<  u.  ~~  )
4 enssdom 7095 . . 3  |-  ~~  C_  ~<_
5 undif 3672 . . 3  |-  (  ~~  C_  ~<_  <-> 
(  ~~  u.  (  ~<_  \ 
~~  ) )  =  ~<_  )
64, 5mpbi 200 . 2  |-  (  ~~  u.  (  ~<_  \  ~~  )
)  =  ~<_
72, 3, 63eqtr3ri 2437 1  |-  ~<_  =  ( 
~<  u.  ~~  )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    \ cdif 3281    u. cun 3282    C_ wss 3284    ~~ cen 7069    ~<_ cdom 7070    ~< csdm 7071
This theorem is referenced by:  brdom2  7100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-opab 4231  df-xp 4847  df-rel 4848  df-f1o 5424  df-en 7073  df-dom 7074  df-sdom 7075
  Copyright terms: Public domain W3C validator