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Theorem dfsymrels2 36282
Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 36254. (Contributed by Peter Mazsa, 20-Jul-2019.)
Assertion
Ref Expression
dfsymrels2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}

Proof of Theorem dfsymrels2
StepHypRef Expression
1 df-symrels 36280 . 2 SymRels = ( Syms ∩ Rels )
2 df-syms 36279 . 2 Syms = {𝑟(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5187 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3404 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 36242 . . . 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 36231 . . . . . . 7 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3404 . . . . . 6 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 219 . . . . 5 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109cnveqd 5718 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1110, 9sseq12d 3910 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
126, 11syl5bb 286 . 2 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
131, 2, 12abeqinbi 36016 1 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1542  wcel 2114  {crab 3057  Vcvv 3398  cin 3842  wss 3843   class class class wbr 5030   × cxp 5523  ccnv 5524  dom cdm 5525  ran crn 5526   Rels crels 35958   S cssr 35959   Syms csyms 35966   SymRels csymrels 35967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-rels 36226  df-ssr 36239  df-syms 36279  df-symrels 36280
This theorem is referenced by:  dfsymrels3  36283  dfsymrels4  36284  elsymrels2  36290  refsymrels2  36302  refrelsredund4  36368
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