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Theorem relcnvfld 6286
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 𝑅 = 𝑅)

Proof of Theorem relcnvfld
StepHypRef Expression
1 relfld 6281 . 2 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2 unidmrn 6285 . 2 𝑅 = (dom 𝑅 ∪ ran 𝑅)
31, 2eqtr4di 2783 1 (Rel 𝑅 𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  cun 3942   cuni 4909  ccnv 5677  dom cdm 5678  ran crn 5679  Rel wrel 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689
This theorem is referenced by:  cnvps  18573  tsrdir  18599
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