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Theorem relcnvfld 5157
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 5152 . 2  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
2 unidmrn 5156 . 2  |-  U. U. `' R  =  ( dom  R  u.  ran  R
)
31, 2eqtr4di 2228 1  |-  ( Rel 
R  ->  U. U. R  =  U. U. `' R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    u. cun 3127   U.cuni 3807   `'ccnv 4621   dom cdm 4622   ran crn 4623   Rel wrel 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-dm 4632  df-rn 4633
This theorem is referenced by: (None)
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