Theorem discsubclem

Description: Lemma for discsubc . (Contributed by Zhi Wang, 1-Nov-2025)

	Ref	Expression
Hypothesis	discsubc.j	$$⊢J=\left(x\in S,y\in S⟼if\left(x=y,\left\{I\left(x\right)\right\},\varnothing \right)\right)$$
Assertion	discsubclem	$$⊢JFn\left(S\times S\right)$$

Proof

Step	Hyp	Ref	Expression
1		discsubc.j	$$⊢J=\left(x\in S,y\in S⟼if\left(x=y,\left\{I\left(x\right)\right\},\varnothing \right)\right)$$
2		snex	$$⊢\left\{I\left(x\right)\right\}\in \mathrm{V}$$
3		0ex	$$⊢\varnothing \in \mathrm{V}$$
4	2 3	ifex	$$⊢if\left(x=y,\left\{I\left(x\right)\right\},\varnothing \right)\in \mathrm{V}$$
5	1 4	fnmpoi	$$⊢JFn\left(S\times S\right)$$