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Theorem relfld 4896
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 4891 . . . 4  |-  ( Rel 
R  ->  R  C_  ( dom  R  X.  ran  R
) )
2 uniss 3642 . . . 4  |-  ( R 
C_  ( dom  R  X.  ran  R )  ->  U. R  C_  U. ( dom  R  X.  ran  R
) )
3 uniss 3642 . . . 4  |-  ( U. R  C_  U. ( dom 
R  X.  ran  R
)  ->  U. U. R  C_ 
U. U. ( dom  R  X.  ran  R ) )
41, 2, 33syl 17 . . 3  |-  ( Rel 
R  ->  U. U. R  C_ 
U. U. ( dom  R  X.  ran  R ) )
5 unixpss 4499 . . 3  |-  U. U. ( dom  R  X.  ran  R )  C_  ( dom  R  u.  ran  R )
64, 5syl6ss 3020 . 2  |-  ( Rel 
R  ->  U. U. R  C_  ( dom  R  u.  ran  R ) )
7 dmrnssfld 4643 . . 3  |-  ( dom 
R  u.  ran  R
)  C_  U. U. R
87a1i 9 . 2  |-  ( Rel 
R  ->  ( dom  R  u.  ran  R ) 
C_  U. U. R )
96, 8eqssd 3025 1  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    u. cun 2980    C_ wss 2982   U.cuni 3621    X. cxp 4389   dom cdm 4391   ran crn 4392   Rel wrel 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by:  relresfld  4897  relcoi1  4899  unidmrn  4900  relcnvfld  4901  unixpm  4903
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