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Theorem relcnvfld 6233
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 6228 . 2 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2 unidmrn 6232 . 2 𝑅 = (dom 𝑅 ∪ ran 𝑅)
31, 2eqtr4di 2791 1 (Rel 𝑅 𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cun 3909   cuni 4866  ccnv 5633  dom cdm 5634  ran crn 5635  Rel wrel 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  cnvps  18472  tsrdir  18498
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