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Theorem relfld 5524
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 5519 . . . 4  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
2 uniss 4259 . . . 4  |-  ( R 
C_  ( dom  R  X.  ran  R )  ->  U. R  C_  U. ( dom  R  X.  ran  R
) )
3 uniss 4259 . . . 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 5109 . . 3  |-  U. U. ( dom  R  X.  ran  R )  C_  ( dom  R  u.  ran  R )
64, 5syl6ss 3509 . 2  |-  ( Rel 
R  ->  U. U. R  C_  ( dom  R  u.  ran  R ) )
7 dmrnssfld 5252 . . 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 3514 1  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    u. cun 3467    C_ wss 3469   U.cuni 4238    X. cxp 4990   dom cdm 4992   ran crn 4993   Rel wrel 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003
This theorem is referenced by:  relresfld  5525  relcoi1  5527  unidmrn  5528  relcnvfld  5529  unixp  5531  lefld  15702  relexpfld  28385  rtrclreclem.min  28395  dfrtrcl2  28396
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