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Theorem dfsymrels2 38017
Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 37985. (Contributed by Peter Mazsa, 20-Jul-2019.)
Assertion
Ref Expression
dfsymrels2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}

Proof of Theorem dfsymrels2
StepHypRef Expression
1 df-symrels 38015 . 2 SymRels = ( Syms ∩ Rels )
2 df-syms 38014 . 2 Syms = {𝑟(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5319 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3477 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 37973 . . . 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 37962 . . . . . . 7 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3477 . . . . . 6 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 215 . . . . 5 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109cnveqd 5878 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1110, 9sseq12d 4013 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
126, 11bitrid 283 . 2 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
131, 2, 12abeqinbi 37725 1 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wcel 2099  {crab 3429  Vcvv 3471  cin 3946  wss 3947   class class class wbr 5148   × cxp 5676  ccnv 5677  dom cdm 5678  ran crn 5679   Rels crels 37650   S cssr 37651   Syms csyms 37658   SymRels csymrels 37659
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-rels 37957  df-ssr 37970  df-syms 38014  df-symrels 38015
This theorem is referenced by:  dfsymrels3  38018  dfsymrels4  38019  elsymrels2  38025  refsymrels2  38037  refrelsredund4  38104
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