Theorem dfsymrels2 39159

Description: Alternate definition of the class of symmetric relations. Cf. the comment of dfrefrels2 39127. (Contributed by Peter Mazsa, 20-Jul-2019.)

Assertion

Ref	Expression
dfsymrels2	⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}

Proof of Theorem dfsymrels2

Step	Hyp	Ref	Expression
1		df-symrels 39157	. 2 ⊢ SymRels = ( Syms ∩ Rels )
2		df-syms 39156	. 2 ⊢ Syms = {𝑟 ∣ ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		inex1g 5287	. . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
4	3	elv 3468	. . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5		brssr 39115	. . . 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))))
6	4, 5	ax-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 38979	. . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3468	. . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
9	8	biimpi 219	. . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
10	9	cnveqd 5859	. . . 4 ⊢ (𝑟 ∈ Rels → ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) = ◡𝑟)
11	10, 9	sseq12d 3978	. . 3 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟))
12	6, 11	bitrid 286	. 2 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟))
13	1, 2, 12	abeqinbi 38789	1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913   class class class wbr 5110   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   Rels crels 38719   S cssr 38720   Syms csyms 38727   SymRels csymrels 38728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-rels 38974  df-ssr 39112  df-syms 39156  df-symrels 39157

This theorem is referenced by:  dfsymrels3  39160  dfsymrels4  39165  elsymrels2  39171  refsymrels2  39183  refrelsredund4  39250