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Theorem dfsymrels2 35226
 Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 35198. (Contributed by Peter Mazsa, 20-Jul-2019.)
Assertion
Ref Expression
dfsymrels2 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}

Proof of Theorem dfsymrels2
StepHypRef Expression
1 df-symrels 35224 . 2 SymRels = ( Syms ∩ Rels )
2 df-syms 35223 . 2 Syms = {𝑟(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5074 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3414 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 35186 . . . 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 35175 . . . . . . 7 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3414 . . . . . 6 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 208 . . . . 5 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109cnveqd 5590 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
1110, 9sseq12d 3884 . . 3 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
126, 11syl5bb 275 . 2 (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟𝑟))
131, 2, 12abeqinbi 34959 1 SymRels = {𝑟 ∈ Rels ∣ 𝑟𝑟}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   = wceq 1507   ∈ wcel 2050  {crab 3086  Vcvv 3409   ∩ cin 3822   ⊆ wss 3823   class class class wbr 4923   × cxp 5399  ◡ccnv 5400  dom cdm 5401  ran crn 5402   Rels crels 34899   S cssr 34900   Syms csyms 34907   SymRels csymrels 34908 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-xp 5407  df-rel 5408  df-cnv 5409  df-dm 5411  df-rn 5412  df-res 5413  df-rels 35170  df-ssr 35183  df-syms 35223  df-symrels 35224 This theorem is referenced by:  dfsymrels3  35227  dfsymrels4  35228  elsymrels2  35234  refsymrels2  35246  refrelsredund4  35312
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