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Metamath Proof Explorer

Definition df-symrel

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 ) is equivalent to satisfying the symmetric relation predicate, see elsymrelsrel . Alternate definitions are dfsymrel2 and dfsymrel3 . (Contributed by Peter Mazsa, 16-Jul-2021)

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

Detailed syntax breakdown

Step Hyp Ref Expression
0 cR ⊢ 𝑅
1 0 wsymrel ⊢ SymRel 𝑅
2 0 cdm ⊢ dom 𝑅
3 0 crn ⊢ ran 𝑅
4 2 3 cxp ⊢ ( dom 𝑅 × ran 𝑅 )
5 0 4 cin ⊢ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) )
6 5 ccnv ⊢ ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) )
7 6 5 wss ⊢ ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) )
8 0 wrel ⊢ Rel 𝑅
9 7 8 wa ⊢ ( ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 )
10 1 9 wb ⊢ ( SymRel 𝑅 ↔ ( ◡ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ⊆ ( 𝑅 ∩ ( dom 𝑅 × ran 𝑅 ) ) ∧ Rel 𝑅 ) )