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Metamath Proof Explorer

Theorem eliniseg

Description: Membership in the inverse image of a singleton. An application is to express initial segments for an order relation. See for example Definition 6.21 of TakeutiZaring p. 30. (Contributed by NM, 28-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

Ref Expression
Hypothesis eliniseg.1 ⊢ C ∈ V
Assertion eliniseg ⊢ B ∈ V → C ∈ A -1 B ↔ C A B

Proof

Step Hyp Ref Expression
1 eliniseg.1 ⊢ C ∈ V
2 elinisegg ⊢ B ∈ V ∧ C ∈ V → C ∈ A -1 B ↔ C A B
3 1 2 mpan2 ⊢ B ∈ V → C ∈ A -1 B ↔ C A B