Theorem bnj97

Description: Technical lemma for bnj150 . 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
Hypothesis	bnj96.1	⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
Assertion	bnj97	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj96.1	⊢ 𝐹 = { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
2		bnj93	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V )
3		0ex	⊢ ∅ ∈ V
4	3	bnj519	⊢ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V → Fun { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ } )
5	1	funeqi	⊢ ( Fun 𝐹 ↔ Fun { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ } )
6	4 5	sylibr	⊢ ( pred ( 𝑥 , 𝐴 , 𝑅 ) ∈ V → Fun 𝐹 )
7	2 6	syl	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → Fun 𝐹 )
8		opex	⊢ ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ ∈ V
9	8	snid	⊢ ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ ∈ { ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ }
10	9 1	eleqtrri	⊢ ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ ∈ 𝐹
11		funopfv	⊢ ( Fun 𝐹 → ( ⟨ ∅ , pred ( 𝑥 , 𝐴 , 𝑅 ) ⟩ ∈ 𝐹 → ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) ) )
12	7 10 11	mpisyl	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ ∅ ) = pred ( 𝑥 , 𝐴 , 𝑅 ) )