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Theorem elsymrels2 34660
Description: Element of the class of symmetric relations. (Contributed by Peter Mazsa, 17-Aug-2021.)
Assertion
Ref Expression
elsymrels2 (𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))

Proof of Theorem elsymrels2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dfsymrels2 34652 . 2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
2 cnveq 5464 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
3 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
42, 3sseq12d 3794 . 2 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
51, 4rabeqel 34386 1 (𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wcel 2155  wss 3732  ccnv 5276   Rels crels 34338   SymRels csymrels 34347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-rels 34596  df-ssr 34609  df-syms 34649  df-symrels 34650
This theorem is referenced by:  elsymrelsrel  34664
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