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Theorem relcnvfld 6182
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 6177 . 2 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2 unidmrn 6181 . 2 𝑅 = (dom 𝑅 ∪ ran 𝑅)
31, 2eqtr4di 2798 1 (Rel 𝑅 𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cun 3890   cuni 4845  ccnv 5589  dom cdm 5590  ran crn 5591  Rel wrel 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-cnv 5598  df-dm 5600  df-rn 5601
This theorem is referenced by:  cnvps  18294  tsrdir  18320
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