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Theorem relfld 5516
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 5511 . . . 4  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
2 uniss 4256 . . . 4  |-  ( R 
C_  ( dom  R  X.  ran  R )  ->  U. R  C_  U. ( dom  R  X.  ran  R
) )
3 uniss 4256 . . . 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 5106 . . 3  |-  U. U. ( dom  R  X.  ran  R )  C_  ( dom  R  u.  ran  R )
64, 5syl6ss 3501 . 2  |-  ( Rel 
R  ->  U. U. R  C_  ( dom  R  u.  ran  R ) )
7 dmrnssfld 5250 . . 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 3506 1  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    u. cun 3459    C_ wss 3461   U.cuni 4235    X. cxp 4986   dom cdm 4988   ran crn 4989   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999
This theorem is referenced by:  relresfld  5517  relcoi1  5519  unidmrn  5520  relcnvfld  5521  unixp  5523  relexp0  12940  relexpfld  12964  rtrclreclem4  12976  dfrtrcl2  12977  lefld  16055
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