Theorem bnj93

Description: Technical lemma for bnj97 . 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
Assertion	bnj93	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		df-bnj15	⊢ ( 𝑅 FrSe 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴 ) )
2	1	simprbi	⊢ ( 𝑅 FrSe 𝐴 → 𝑅 Se 𝐴 )
3		df-bnj13	⊢ ( 𝑅 Se 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
4	2 3	sylib	⊢ ( 𝑅 FrSe 𝐴 → ∀ 𝑥 ∈ 𝐴 pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
5	4	r19.21bi	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )