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Theorem relfld 5358
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 5353 . . . 4  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
2 uniss 4107 . . . 4  |-  ( R 
C_  ( dom  R  X.  ran  R )  ->  U. R  C_  U. ( dom  R  X.  ran  R
) )
3 uniss 4107 . . . 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 4950 . . 3  |-  U. U. ( dom  R  X.  ran  R )  C_  ( dom  R  u.  ran  R )
64, 5syl6ss 3363 . 2  |-  ( Rel 
R  ->  U. U. R  C_  ( dom  R  u.  ran  R ) )
7 dmrnssfld 5093 . . 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 3368 1  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    u. cun 3321    C_ wss 3323   U.cuni 4086    X. cxp 4833   dom cdm 4835   ran crn 4836   Rel wrel 4840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-dm 4845  df-rn 4846
This theorem is referenced by:  relresfld  5359  relcoi1  5361  unidmrn  5362  relcnvfld  5363  unixp  5365  lefld  15388  relexpfld  27290  rtrclreclem.min  27300  dfrtrcl2  27301
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