Metamath Proof Explorer

Theorem bnj911

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj911.1	$⊢ φ \leftrightarrow f (\emptyset) = pred (X, A, R)$
	bnj911.2	$⊢ ψ \leftrightarrow \forall i \in ω (suc i \in n \to f (suc i) = ⋃_{y \in f (i)} pred (y, A, R))$
Assertion	bnj911	$⊢ (f Fn n \land φ \land ψ) \to \forall i (f Fn n \land φ \land ψ)$

Proof

Step	Hyp	Ref	Expression
1		bnj911.1	$⊢ φ \leftrightarrow f (\emptyset) = pred (X, A, R)$
2		bnj911.2	$⊢ ψ \leftrightarrow \forall i \in ω (suc i \in n \to f (suc i) = ⋃_{y \in f (i)} pred (y, A, R))$
3	2	bnj1095	$⊢ ψ \to \forall i ψ$
4	3	bnj1350	$⊢ (f Fn n \land φ \land ψ) \to \forall i (f Fn n \land φ \land ψ)$