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

Theorem dfdom2 7125
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 7104 . . 3  |-  ~<  =  (  ~<_  \  ~~  )
21uneq2i 3490 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~~  u.  (  ~<_  \  ~~  ) )
3 uncom 3483 . 2  |-  (  ~~  u.  ~<  )  =  ( 
~<  u.  ~~  )
4 enssdom 7124 . . 3  |-  ~~  C_  ~<_
5 undif 3700 . . 3  |-  (  ~~  C_  ~<_  <-> 
(  ~~  u.  (  ~<_  \ 
~~  ) )  =  ~<_  )
64, 5mpbi 200 . 2  |-  (  ~~  u.  (  ~<_  \  ~~  )
)  =  ~<_
72, 3, 63eqtr3ri 2464 1  |-  ~<_  =  ( 
~<  u.  ~~  )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    \ cdif 3309    u. cun 3310    C_ wss 3312    ~~ cen 7098    ~<_ cdom 7099    ~< csdm 7100
This theorem is referenced by:  brdom2  7129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-rel 4877  df-f1o 5453  df-en 7102  df-dom 7103  df-sdom 7104
  Copyright terms: Public domain W3C validator