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Metamath Proof Explorer

Definition df-reflexive

Description: Define reflexive relation; relation R is reflexive over the set A iff A. x e. A x R x . (Contributed by David A. Wheeler, 1-Dec-2019)

Ref Expression
Assertion df-reflexive ( 𝑅 Reflexive 𝐴 ↔ ( 𝑅 ⊆ ( 𝐴 × 𝐴 ) ∧ ∀ 𝑥 ∈ 𝐴 𝑥 𝑅 𝑥 ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cR ⊢ 𝑅
1 cA ⊢ 𝐴
2 1 0 wreflexive ⊢ 𝑅 Reflexive 𝐴
3 1 1 cxp ⊢ ( 𝐴 × 𝐴 )
4 0 3 wss ⊢ 𝑅 ⊆ ( 𝐴 × 𝐴 )
5 vx ⊢ 𝑥
6 5 cv ⊢ 𝑥
7 6 6 0 wbr ⊢ 𝑥 𝑅 𝑥
8 7 5 1 wral ⊢ ∀ 𝑥 ∈ 𝐴 𝑥 𝑅 𝑥
9 4 8 wa ⊢ ( 𝑅 ⊆ ( 𝐴 × 𝐴 ) ∧ ∀ 𝑥 ∈ 𝐴 𝑥 𝑅 𝑥 )
10 2 9 wb ⊢ ( 𝑅 Reflexive 𝐴 ↔ ( 𝑅 ⊆ ( 𝐴 × 𝐴 ) ∧ ∀ 𝑥 ∈ 𝐴 𝑥 𝑅 𝑥 ) )