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Theorem elrels6 37871
Description: Equivalent expressions for an element of the relations class. (Contributed by Peter Mazsa, 21-Jul-2021.)
Assertion
Ref Expression
elrels6 (𝑅𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅))

Proof of Theorem elrels6
StepHypRef Expression
1 elrelsrel 37868 . 2 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2 dfrel6 37727 . 2 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
31, 2bitrdi 287 1 (𝑅𝑉 → (𝑅 ∈ Rels ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  cin 3942   × cxp 5667  dom cdm 5669  ran crn 5670  Rel wrel 5674   Rels crels 37556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-rels 37866
This theorem is referenced by:  dfrefrels2  37894  dfcnvrefrels2  37909  dfsymrels2  37926  dftrrels2  37956
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