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Theorem relcnvfld 5373
Description: if  R is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.)
Assertion
Ref Expression
relcnvfld  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)

Proof of Theorem relcnvfld
StepHypRef Expression
1 relfld 5368 . 2  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
2 unidmrn 5372 . 2  |-  U. U. `' R  =  ( dom  R  u.  ran  R
)
31, 2syl6eqr 2493 1  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    u. cun 3331   U.cuni 4096   `'ccnv 4844   dom cdm 4845   ran crn 4846   Rel wrel 4850
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 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856
This theorem is referenced by:  cnvps  15387  tsrdir  15413  relexpcnv  27340
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