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Theorem relcnvfld 6112
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 6107 . 2 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2 unidmrn 6111 . 2 𝑅 = (dom 𝑅 ∪ ran 𝑅)
31, 2syl6eqr 2877 1 (Rel 𝑅 𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cun 3916   cuni 4819  ccnv 5535  dom cdm 5536  ran crn 5537  Rel wrel 5541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-cnv 5544  df-dm 5546  df-rn 5547
This theorem is referenced by:  cnvps  17811  tsrdir  17837
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