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Theorem dfsymrel2 38550
Description: Alternate definition of the symmetric relation predicate. (Contributed by Peter Mazsa, 19-Apr-2019.) (Revised by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
dfsymrel2 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))

Proof of Theorem dfsymrel2
StepHypRef Expression
1 df-symrel 38545 . 2 ( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 dfrel6 38348 . . . . . 6 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32biimpi 216 . . . . 5 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
43cnveqd 5886 . . . 4 (Rel 𝑅(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
54, 3sseq12d 4017 . . 3 (Rel 𝑅 → ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ↔ 𝑅𝑅))
65pm5.32ri 575 . 2 (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (𝑅𝑅 ∧ Rel 𝑅))
71, 6bitri 275 1 ( SymRel 𝑅 ↔ (𝑅𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  cin 3950  wss 3951   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  Rel wrel 5690   SymRel wsymrel 38194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-symrel 38545
This theorem is referenced by:  dfsymrel3  38551  dfsymrel4  38552  dfsymrel5  38553  elsymrelsrel  38558  symreleq  38559  symrelcoss  38561  refsymrel2  38568  eqvrelsym  38606  refrelredund4  38636
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