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Theorem relcnvfld 5084
 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 5079 . 2 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2 unidmrn 5083 . 2 𝑅 = (dom 𝑅 ∪ ran 𝑅)
31, 2eqtr4di 2192 1 (Rel 𝑅 𝑅 = 𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∪ cun 3076  ∪ cuni 3746  ◡ccnv 4550  dom cdm 4551  ran crn 4552  Rel wrel 4556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-br 3940  df-opab 4000  df-xp 4557  df-rel 4558  df-cnv 4559  df-dm 4561  df-rn 4562 This theorem is referenced by: (None)
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