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Theorem elsymrels2 36229
 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 36221 . 2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
2 cnveq 5713 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
3 id 22 . . 3 (𝑟 = 𝑅𝑟 = 𝑅)
42, 3sseq12d 3925 . 2 (𝑟 = 𝑅 → (𝑟𝑟𝑅𝑅))
51, 4rabeqel 35956 1 (𝑅 ∈ SymRels ↔ (𝑅𝑅𝑅 ∈ Rels ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3858  ◡ccnv 5523   Rels crels 35895   SymRels csymrels 35904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-rel 5531  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536  df-rels 36165  df-ssr 36178  df-syms 36218  df-symrels 36219 This theorem is referenced by:  elsymrelsrel  36233
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