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 ∣ ◡𝑟 ⊆ 𝑟}

Proof of Theorem dfsymrels2

Step	Hyp	Ref	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)
4	3	elv 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))))
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 38018	. . . . . . 7 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3469	. . . . . 6 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
9	8	biimpi 215	. . . . 5 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
10	9	cnveqd 5872	. . . 4 ⊢ (𝑟 ∈ Rels → ◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) = ◡𝑟)
11	10, 9	sseq12d 4006	. . 3 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟))
12	6, 11	bitrid 282	. 2 ⊢ (𝑟 ∈ Rels → (◡(𝑟 ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ◡𝑟 ⊆ 𝑟))
13	1, 2, 12	abeqinbi 37781	1 ⊢ SymRels = {𝑟 ∈ Rels ∣ ◡𝑟 ⊆ 𝑟}

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