Theorem frgrncvvdeqlem4

Description: Lemma 4 for frgrncvvdeq . The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-Dec-2017) (Revised by AV, 10-May-2021)

	Ref	Expression
Hypotheses	frgrncvvdeq.v1	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	frgrncvvdeq.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
	frgrncvvdeq.nx	$$⊢D=G\mathrm{NeighbVtx}X$$
	frgrncvvdeq.ny	$$⊢N=G\mathrm{NeighbVtx}Y$$
	frgrncvvdeq.x	$$⊢\phi \to X\in V$$
	frgrncvvdeq.y	$$⊢\phi \to Y\in V$$
	frgrncvvdeq.ne	$$⊢\phi \to X\ne Y$$
	frgrncvvdeq.xy	$$⊢\phi \to Y\notin D$$
	frgrncvvdeq.f	$$⊢\phi \to G\in \mathrm{FriendGraph}$$
	frgrncvvdeq.a	$$⊢A=\left(x\in D⟼\left(\iota y\in N|\left\{x,y\right\}\in E\right)\right)$$
Assertion	frgrncvvdeqlem4	$$⊢\phi \to A:D⟶N$$

Proof

Step	Hyp	Ref	Expression
1		frgrncvvdeq.v1	$$⊢V=\mathrm{Vtx}\left(G\right)$$
2		frgrncvvdeq.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
3		frgrncvvdeq.nx	$$⊢D=G\mathrm{NeighbVtx}X$$
4		frgrncvvdeq.ny	$$⊢N=G\mathrm{NeighbVtx}Y$$
5		frgrncvvdeq.x	$$⊢\phi \to X\in V$$
6		frgrncvvdeq.y	$$⊢\phi \to Y\in V$$
7		frgrncvvdeq.ne	$$⊢\phi \to X\ne Y$$
8		frgrncvvdeq.xy	$$⊢\phi \to Y\notin D$$
9		frgrncvvdeq.f	$$⊢\phi \to G\in \mathrm{FriendGraph}$$
10		frgrncvvdeq.a	$$⊢A=\left(x\in D⟼\left(\iota y\in N|\left\{x,y\right\}\in E\right)\right)$$
11	1 2 3 4 5 6 7 8 9 10	frgrncvvdeqlem2	$$⊢\left(\phi \wedge x\in D\right)\to \exists !y\in N\left\{x,y\right\}\in E$$
12		riotacl	$$⊢\exists !y\in N\left\{x,y\right\}\in E\to \left(\iota y\in N|\left\{x,y\right\}\in E\right)\in N$$
13	11 12	syl	$$⊢\left(\phi \wedge x\in D\right)\to \left(\iota y\in N|\left\{x,y\right\}\in E\right)\in N$$
14	13 10	fmptd	$$⊢\phi \to A:D⟶N$$