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Theorem relcnvfld 5538
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 5533 . 2  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
2 unidmrn 5537 . 2  |-  U. U. `' R  =  ( dom  R  u.  ran  R
)
31, 2syl6eqr 2526 1  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    u. cun 3474   U.cuni 4245   `'ccnv 4998   dom cdm 4999   ran crn 5000   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  cnvps  15699  tsrdir  15725  relexpcnv  28559
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