Theorem bnj96

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) (Revised by Mario Carneiro, 6-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj96.1	$$⊢F=\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
Assertion	bnj96	$$⊢\left(RFrSeA\wedge x\in A\right)\to \mathrm{dom}F=1_{𝑜}$$

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		dmsnopg	$$⊢pred\left(x,A,R\right)\in \mathrm{V}\to \mathrm{dom}\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}=\left\{\varnothing \right\}$$
4	2 3	syl	$$⊢\left(RFrSeA\wedge x\in A\right)\to \mathrm{dom}\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}=\left\{\varnothing \right\}$$
5	1	dmeqi	$$⊢\mathrm{dom}F=\mathrm{dom}\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
6		df1o2	$$⊢1_{𝑜}=\left\{\varnothing \right\}$$
7	4 5 6	3eqtr4g	$$⊢\left(RFrSeA\wedge x\in A\right)\to \mathrm{dom}F=1_{𝑜}$$