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Definition df-symrel 35647
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 35646) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel 35660. Alternate definitions are dfsymrel2 35652 and dfsymrel3 35653. (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
StepHypRef Expression
1 cR . . 3 class 𝑅
21wsymrel 35333 . 2 wff SymRel 𝑅
31cdm 5553 . . . . . . 7 class dom 𝑅
41crn 5554 . . . . . . 7 class ran 𝑅
53, 4cxp 5551 . . . . . 6 class (dom 𝑅 × ran 𝑅)
61, 5cin 3938 . . . . 5 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
76ccnv 5552 . . . 4 class (𝑅 ∩ (dom 𝑅 × ran 𝑅))
87, 6wss 3939 . . 3 wff (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅))
91wrel 5558 . . 3 wff Rel 𝑅
108, 9wa 396 . 2 wff ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅)
112, 10wb 207 1 wff ( SymRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ (𝑅 ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
Colors of variables: wff setvar class
This definition is referenced by:  dfsymrel2  35652
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