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Theorem dfsymrels2 36659
Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 36631. (Contributed by Peter Mazsa, 20-Jul-2019.)
Assertion
Ref Expression
dfsymrels2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}

Proof of Theorem dfsymrels2
StepHypRef Expression
1 df-symrels 36657 . 2 SymRels = ( Syms ∩ Rels )
2 df-syms 36656 . 2 Syms = {𝑟(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5243 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3438 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 36619 . . . 4 ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟))))
64, 5ax-mp 5 . . 3 ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)))
7 elrels6 36608 . . . . . . 7 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3438 . . . . . 6 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 215 . . . . 5 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109cnveqd 5784 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1110, 9sseq12d 3954 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
126, 11syl5bb 283 . 2 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
131, 2, 12abeqinbi 36393 1 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1539  wcel 2106  {crab 3068  Vcvv 3432  cin 3886  wss 3887   class class class wbr 5074   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590   Rels crels 36335   S cssr 36336   Syms csyms 36343   SymRels csymrels 36344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-rels 36603  df-ssr 36616  df-syms 36656  df-symrels 36657
This theorem is referenced by:  dfsymrels3  36660  dfsymrels4  36661  elsymrels2  36667  refsymrels2  36679  refrelsredund4  36745
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