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Theorem dfsymrels2 38073
Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 38041. (Contributed by Peter Mazsa, 20-Jul-2019.)
Assertion
Ref Expression
dfsymrels2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}

Proof of Theorem dfsymrels2
StepHypRef Expression
1 df-symrels 38071 . 2 SymRels = ( Syms ∩ Rels )
2 df-syms 38070 . 2 Syms = {𝑟(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5314 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3469 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 38029 . . . 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 38018 . . . . . . 7 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3469 . . . . . 6 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 215 . . . . 5 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109cnveqd 5872 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1110, 9sseq12d 4006 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
126, 11bitrid 282 . 2 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
131, 2, 12abeqinbi 37781 1 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  {crab 3419  Vcvv 3463  cin 3938  wss 3939   class class class wbr 5143   × cxp 5670  ccnv 5671  dom cdm 5672  ran crn 5673   Rels crels 37707   S cssr 37708   Syms csyms 37715   SymRels csymrels 37716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-rels 38013  df-ssr 38026  df-syms 38070  df-symrels 38071
This theorem is referenced by:  dfsymrels3  38074  dfsymrels4  38075  elsymrels2  38081  refsymrels2  38093  refrelsredund4  38160
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