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Theorem dfsymrel2 38531
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 38526 . 2 ( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 dfrel6 38329 . . . . . 6 (Rel 𝑅 ↔ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
32biimpi 216 . . . . 5 (Rel 𝑅 → (𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
43cnveqd 5889 . . . 4 (Rel 𝑅(𝑅 ∩ (dom 𝑅 × ran 𝑅)) = 𝑅)
54, 3sseq12d 4029 . . 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 1537  cin 3962  wss 3963   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  Rel wrel 5694   SymRel wsymrel 38174
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  df-symrel 38526
This theorem is referenced by:  dfsymrel3  38532  dfsymrel4  38533  dfsymrel5  38534  elsymrelsrel  38539  symreleq  38540  symrelcoss  38542  refsymrel2  38549  eqvrelsym  38587  refrelredund4  38617
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