Definition df-symrel 38545

Description: Define the symmetric relation predicate. (Read: 𝑅 is a symmetric relation.) For sets, being an element of the class of symmetric relations (df-symrels 38544) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 38558. Alternate definitions are dfsymrel2 38550 and dfsymrel3 38551. (Contributed by Peter Mazsa, 16-Jul-2021.)

Assertion

Ref	Expression
df-symrel	⊢ ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

Detailed syntax breakdown of Definition df-symrel

Step	Hyp	Ref	Expression
1		cR	. . 3 class 𝑅
2	1	wsymrel 38194	. 2 wff SymRel 𝑅
3	1	cdm 5685	. . . . . . 7 class dom 𝑅
4	1	crn 5686	. . . . . . 7 class ran 𝑅
5	3, 4	cxp 5683	. . . . . 6 class (dom 𝑅 × ran 𝑅)
6	1, 5	cin 3950	. . . . 5 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
7	6	ccnv 5684	. . . 4 class ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅))
8	7, 6	wss 3951	. . 3 wff ◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅))
9	1	wrel 5690	. . 3 wff Rel 𝑅
10	8, 9	wa 395	. 2 wff (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
11	2, 10	wb 206	1 wff ( SymRel 𝑅 ↔ (◡(𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))

This definition is referenced by:  dfsymrel2  38550