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Theorem dfrel6 38329
Description: Alternate definition of the relation predicate. (Contributed by Peter Mazsa, 14-Mar-2019.)
Assertion
Ref Expression
dfrel6 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)

Proof of Theorem dfrel6
StepHypRef Expression
1 dfrel5 38328 . 2 (Rel 𝑅 ↔ (𝑅 ↾ dom 𝑅) = 𝑅)
2 dfres3 6005 . . 3 (𝑅 ↾ dom 𝑅) = (𝑅 ∩ (dom 𝑅 × ran 𝑅))
32eqeq1i 2740 . 2 ((𝑅 ↾ dom 𝑅) = 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
41, 3bitri 275 1 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  cin 3962   × cxp 5687  dom cdm 5689  ran crn 5690  cres 5691  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by:  cnvref4  38332  elrels6  38472  dfrefrel2  38497  dfcnvrefrel2  38512  dfsymrel2  38531  dftrrel2  38559
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