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	$$⊢F=\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
Assertion	bnj97	$$⊢\left(RFrSeA\wedge x\in A\right)\to F\left(\varnothing \right)=pred\left(x,A,R\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj96.1	$$⊢F=\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
2		bnj93	$$⊢\left(RFrSeA\wedge x\in A\right)\to pred\left(x,A,R\right)\in \mathrm{V}$$
3		0ex	$$⊢\varnothing \in \mathrm{V}$$
4	3	bnj519	$$⊢pred\left(x,A,R\right)\in \mathrm{V}\to \mathrm{Fun}\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
5	1	funeqi	$$⊢\mathrm{Fun}F\leftrightarrow \mathrm{Fun}\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
6	4 5	sylibr	$$⊢pred\left(x,A,R\right)\in \mathrm{V}\to \mathrm{Fun}F$$
7	2 6	syl	$$⊢\left(RFrSeA\wedge x\in A\right)\to \mathrm{Fun}F$$
8		opex	$$⊢⟨\varnothing ,pred\left(x,A,R\right)⟩\in \mathrm{V}$$
9	8	snid	$$⊢⟨\varnothing ,pred\left(x,A,R\right)⟩\in \left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
10	9 1	eleqtrri	$$⊢⟨\varnothing ,pred\left(x,A,R\right)⟩\in F$$
11		funopfv	$$⊢\mathrm{Fun}F\to \left(⟨\varnothing ,pred\left(x,A,R\right)⟩\in F\to F\left(\varnothing \right)=pred\left(x,A,R\right)\right)$$
12	7 10 11	mpisyl	$$⊢\left(RFrSeA\wedge x\in A\right)\to F\left(\varnothing \right)=pred\left(x,A,R\right)$$