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Theorem elrels6 37966
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 37963 . 2 (𝑅𝑉 → (𝑅 ∈ Rels ↔ Rel 𝑅))
2 dfrel6 37823 . 2 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
31, 2bitrdi 286 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 3946   × cxp 5678  dom cdm 5680  ran crn 5681  Rel wrel 5685   Rels crels 37655
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-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-dm 5690  df-rn 5691  df-res 5692  df-rels 37961
This theorem is referenced by:  dfrefrels2  37989  dfcnvrefrels2  38004  dfsymrels2  38021  dftrrels2  38051
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