Theorem grlimpredg

Description: For two locally isomorphic graphs G and H and a vertex A of G there is a bijection f mapping the closed neighborhood N of A onto the closed neighborhood M of ( FA ) , so that the mapped vertices of an edge { A , B } containing the vertex A is an edge in H . (Contributed by AV, 27-Dec-2025)

	Ref	Expression
Hypotheses	clnbgrvtxedg.n	|- N = ( G ClNeighbVtx A )
	clnbgrvtxedg.i	|- I = ( Edg ` G )
	clnbgrvtxedg.k	|- K = { x e. I | x C_ N }
	grlimedgclnbgr.m	|- M = ( H ClNeighbVtx ( F ` A ) )
	grlimedgclnbgr.j	|- J = ( Edg ` H )
	grlimedgclnbgr.l	|- L = { x e. J | x C_ M }
Assertion	grlimpredg	|- ( ( ( G e. USPGraph /\ H e. USPGraph ) /\ F e. ( G GraphLocIso H ) /\ ( A e. V /\ B e. W /\ { A , B } e. I ) ) -> E. f ( f : N -1-1-onto-> M /\ { ( f ` A ) , ( f ` B ) } e. J ) )

Proof

Step	Hyp	Ref	Expression
1		clnbgrvtxedg.n	|- N = ( G ClNeighbVtx A )
2		clnbgrvtxedg.i	|- I = ( Edg ` G )
3		clnbgrvtxedg.k	|- K = { x e. I | x C_ N }
4		grlimedgclnbgr.m	|- M = ( H ClNeighbVtx ( F ` A ) )
5		grlimedgclnbgr.j	|- J = ( Edg ` H )
6		grlimedgclnbgr.l	|- L = { x e. J | x C_ M }
7	1 2 3 4 5 6	grlimprclnbgredg	|- ( ( ( G e. USPGraph /\ H e. USPGraph ) /\ F e. ( G GraphLocIso H ) /\ ( A e. V /\ B e. W /\ { A , B } e. I ) ) -> E. f ( f : N -1-1-onto-> M /\ { ( f ` A ) , ( f ` B ) } e. L ) )
8		sseq1	|- ( x = { ( f ` A ) , ( f ` B ) } -> ( x C_ M <-> { ( f ` A ) , ( f ` B ) } C_ M ) )
9	8 6	elrab2	|- ( { ( f ` A ) , ( f ` B ) } e. L <-> ( { ( f ` A ) , ( f ` B ) } e. J /\ { ( f ` A ) , ( f ` B ) } C_ M ) )
10		simpl	|- ( ( { ( f ` A ) , ( f ` B ) } e. J /\ { ( f ` A ) , ( f ` B ) } C_ M ) -> { ( f ` A ) , ( f ` B ) } e. J )
11	10	a1i	|- ( ( ( ( G e. USPGraph /\ H e. USPGraph ) /\ F e. ( G GraphLocIso H ) /\ ( A e. V /\ B e. W /\ { A , B } e. I ) ) /\ f : N -1-1-onto-> M ) -> ( ( { ( f ` A ) , ( f ` B ) } e. J /\ { ( f ` A ) , ( f ` B ) } C_ M ) -> { ( f ` A ) , ( f ` B ) } e. J ) )
12	9 11	biimtrid	|- ( ( ( ( G e. USPGraph /\ H e. USPGraph ) /\ F e. ( G GraphLocIso H ) /\ ( A e. V /\ B e. W /\ { A , B } e. I ) ) /\ f : N -1-1-onto-> M ) -> ( { ( f ` A ) , ( f ` B ) } e. L -> { ( f ` A ) , ( f ` B ) } e. J ) )
13	12	imdistanda	|- ( ( ( G e. USPGraph /\ H e. USPGraph ) /\ F e. ( G GraphLocIso H ) /\ ( A e. V /\ B e. W /\ { A , B } e. I ) ) -> ( ( f : N -1-1-onto-> M /\ { ( f ` A ) , ( f ` B ) } e. L ) -> ( f : N -1-1-onto-> M /\ { ( f ` A ) , ( f ` B ) } e. J ) ) )
14	13	eximdv	|- ( ( ( G e. USPGraph /\ H e. USPGraph ) /\ F e. ( G GraphLocIso H ) /\ ( A e. V /\ B e. W /\ { A , B } e. I ) ) -> ( E. f ( f : N -1-1-onto-> M /\ { ( f ` A ) , ( f ` B ) } e. L ) -> E. f ( f : N -1-1-onto-> M /\ { ( f ` A ) , ( f ` B ) } e. J ) ) )
15	7 14	mpd	|- ( ( ( G e. USPGraph /\ H e. USPGraph ) /\ F e. ( G GraphLocIso H ) /\ ( A e. V /\ B e. W /\ { A , B } e. I ) ) -> E. f ( f : N -1-1-onto-> M /\ { ( f ` A ) , ( f ` B ) } e. J ) )