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

Theorem relfld 5464
Description: The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
relfld  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )

Proof of Theorem relfld
StepHypRef Expression
1 relssdmrn 5459 . . . 4  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
2 uniss 4213 . . . 4  |-  ( R 
C_  ( dom  R  X.  ran  R )  ->  U. R  C_  U. ( dom  R  X.  ran  R
) )
3 uniss 4213 . . . 4  |-  ( U. R  C_  U. ( dom 
R  X.  ran  R
)  ->  U. U. R  C_ 
U. U. ( dom  R  X.  ran  R ) )
41, 2, 33syl 20 . . 3  |-  ( Rel 
R  ->  U. U. R  C_ 
U. U. ( dom  R  X.  ran  R ) )
5 unixpss 5056 . . 3  |-  U. U. ( dom  R  X.  ran  R )  C_  ( dom  R  u.  ran  R )
64, 5syl6ss 3469 . 2  |-  ( Rel 
R  ->  U. U. R  C_  ( dom  R  u.  ran  R ) )
7 dmrnssfld 5199 . . 3  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
87a1i 11 . 2  |-  ( Rel 
R  ->  ( dom  R  u.  ran  R ) 
C_  U. U. R )
96, 8eqssd 3474 1  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    u. cun 3427    C_ wss 3429   U.cuni 4192    X. cxp 4939   dom cdm 4941   ran crn 4942   Rel wrel 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-cnv 4949  df-dm 4951  df-rn 4952
This theorem is referenced by:  relresfld  5465  relcoi1  5467  unidmrn  5468  relcnvfld  5469  unixp  5471  lefld  15507  relexpfld  27476  rtrclreclem.min  27486  dfrtrcl2  27487
  Copyright terms: Public domain W3C validator