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Theorem elsymrels2 38251
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 38243 . 2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
2 cnveq 5880 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
3 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
42, 3sseq12d 4013 . 2 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
51, 4rabeqel 37952 1 (𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  wss 3947  ccnv 5681   Rels crels 37878   SymRels csymrels 37887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-rels 38183  df-ssr 38196  df-syms 38240  df-symrels 38241
This theorem is referenced by:  elsymrelsrel  38255
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