Theorem grlimprclnbgrvtx 48497

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 (𝐹‘𝐴), so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀 containing the vertex (𝐹‘𝐴). (Contributed by AV, 28-Dec-2025.)

Hypotheses

Ref	Expression
clnbgrvtxedg.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
clnbgrvtxedg.i	⊢ 𝐼 = (Edg‘𝐺)
clnbgrvtxedg.k	⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
grlimedgclnbgr.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))
grlimedgclnbgr.j	⊢ 𝐽 = (Edg‘𝐻)
grlimedgclnbgr.l	⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}

Assertion

Ref	Expression
grlimprclnbgrvtx	⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))

Distinct variable groups:   𝑥,𝐼   𝑥,𝑁   𝑥,𝐴,𝑓   𝑓,𝐹,𝑥   𝑓,𝐺,𝑥   𝑓,𝐻,𝑥   𝑓,𝐼   𝑥,𝐽   𝐵,𝑓,𝑥   𝑓,𝑉   𝑓,𝑊   𝑥,𝑀   𝑥,𝑓

Allowed substitution hints:   𝐽(𝑓)   𝐾(𝑥,𝑓)   𝐿(𝑥,𝑓)   𝑀(𝑓)   𝑁(𝑓)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem grlimprclnbgrvtx

Step	Hyp	Ref	Expression
1		clnbgrvtxedg.n	. . 3 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
2		clnbgrvtxedg.i	. . 3 ⊢ 𝐼 = (Edg‘𝐺)
3		clnbgrvtxedg.k	. . 3 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
4		grlimedgclnbgr.m	. . 3 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))
5		grlimedgclnbgr.j	. . 3 ⊢ 𝐽 = (Edg‘𝐻)
6		grlimedgclnbgr.l	. . 3 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
7	1, 2, 3, 4, 5, 6	grlimprclnbgredg 48495	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
8		simprl 776	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → 𝑓:𝑁–1-1-onto→𝑀)
9		sseq1 3947	. . . . . . . . 9 ⊢ (𝑥 = {(𝑓‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ 𝑀 ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
10	9, 6	elrab2 3639	. . . . . . . 8 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
11	10	bilani 505	. . . . . . 7 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
12	11	adantl 482	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
13		fvex 6847	. . . . . . . . 9 ⊢ (𝑓‘𝐴) ∈ V
14		fvex 6847	. . . . . . . . 9 ⊢ (𝑓‘𝐵) ∈ V
15	13, 14	prss 4758	. . . . . . . 8 ⊢ (((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀) ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀)
16		uspgrupgr 29272	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
17	16	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐻 ∈ UPGraph)
18	17	3ad2ant1 1139	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐻 ∈ UPGraph)
19	18	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) → 𝐻 ∈ UPGraph)
20	4	eleq2i 2832	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝐴) ∈ 𝑀 ↔ (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
21	5	clnbupgreli 48333	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ∈ UPGraph ∧ (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽))
22	21	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽)))
23	20, 22	biimtrid 243	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐴) ∈ 𝑀 → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽)))
24	4	eleq2i 2832	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝐵) ∈ 𝑀 ↔ (𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
25	5	clnbupgreli 48333	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ∈ UPGraph ∧ (𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽))
26	25	ex 413	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)))
27	24, 26	biimtrid 243	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐵) ∈ 𝑀 → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)))
28	23, 27	anim12d 615	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ UPGraph → (((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀) → (((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽) ∧ ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽))))
29	19, 28	syl 17	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) → (((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀) → (((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽) ∧ ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽))))
30	29	imp 407	. . . . . . . . . 10 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) → (((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽) ∧ ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)))
31		prcom 4671	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {(𝑓‘𝐴), (𝑓‘𝐵)} = {(𝑓‘𝐵), (𝑓‘𝐴)}
32		preq1 4672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝑓‘𝐵), (𝑓‘𝐴)} = {(𝐹‘𝐴), (𝑓‘𝐴)})
33	31, 32	eqtrid 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝑓‘𝐴), (𝑓‘𝐵)} = {(𝐹‘𝐴), (𝑓‘𝐴)})
34	33	eleq1d 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
35	34	biimpcd 250	. . . . . . . . . . . . . . . . . 18 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
36	35	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
37	36	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
38	37	ad2antrr 732	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
39		prcom 4671	. . . . . . . . . . . . . . . . . . 19 ⊢ {(𝑓‘𝐵), (𝐹‘𝐴)} = {(𝐹‘𝐴), (𝑓‘𝐵)}
40	39	eleq1i 2831	. . . . . . . . . . . . . . . . . 18 ⊢ ({(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)
41	40	bilani 505	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)
42	19	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → 𝐻 ∈ UPGraph)
43		fvex 6847	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹‘𝐴) ∈ V
44	14, 43	pm3.2i 471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝐵) ∈ V ∧ (𝐹‘𝐴) ∈ V)
45	44	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → ((𝑓‘𝐵) ∈ V ∧ (𝐹‘𝐴) ∈ V))
46		simpr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)
47	42, 45, 46	3jca 1134	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → (𝐻 ∈ UPGraph ∧ ((𝑓‘𝐵) ∈ V ∧ (𝐹‘𝐴) ∈ V) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽))
48		eqid 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
49	48, 5	upgrpredgv 29233	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 ∈ UPGraph ∧ ((𝑓‘𝐵) ∈ V ∧ (𝐹‘𝐴) ∈ V) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → ((𝑓‘𝐵) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝐴) ∈ (Vtx‘𝐻)))
50		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝐵) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝐴) ∈ (Vtx‘𝐻)) → (𝐹‘𝐴) ∈ (Vtx‘𝐻))
51	47, 49, 50	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → (𝐹‘𝐴) ∈ (Vtx‘𝐻))
52	48	clnbgrvtxel 48327	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝐴) ∈ (Vtx‘𝐻) → (𝐹‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
53	4	eleq2i 2832	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝐴) ∈ 𝑀 ↔ (𝐹‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
54	52, 53	sylibr 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝐴) ∈ (Vtx‘𝐻) → (𝐹‘𝐴) ∈ 𝑀)
55	51, 54	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → (𝐹‘𝐴) ∈ 𝑀)
56		simplrr 783	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → (𝑓‘𝐵) ∈ 𝑀)
57	55, 56	prssd 4760	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀)
58		sseq1 3947	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ 𝑀 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
59	58, 6	elrab2 3639	. . . . . . . . . . . . . . . . 17 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
60	41, 57, 59	sylanbrc 589	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)
61	60	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) → ({(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽 → {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
62	38, 61	orim12d 972	. . . . . . . . . . . . . 14 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) → (((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → ({(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)))
63	62	imp 407	. . . . . . . . . . . . 13 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)) → ({(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
64	63	orcomd 877	. . . . . . . . . . . 12 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) ∧ ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
65	64	ex 413	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) → (((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
66	65	adantld 491	. . . . . . . . . 10 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) → ((((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽) ∧ ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
67	30, 66	mpd 15	. . . . . . . . 9 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) ∧ ((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀)) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
68	67	ex 413	. . . . . . . 8 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) → (((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
69	15, 68	biimtrrid 244	. . . . . . 7 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀 → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
70	69	expimpd 454	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → (({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
71	12, 70	mpd 15	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
72	8, 71	jca 516	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → (𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
73	72	ex 413	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ((𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → (𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))))
74	73	eximdv 1924	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))))
75	7, 74	mpd 15	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {crab 3392  Vcvv 3432   ⊆ wss 3890  {cpr 4564  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7363  Vtxcvtx 29090  Edgcedg 29141  UPGraphcupgr 29174  USPGraphcuspgr 29242   ClNeighbVtx cclnbgr 48316   GraphLocIso cgrlim 48474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9823  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-n0 12436  df-xnn0 12509  df-z 12523  df-uz 12787  df-fz 13460  df-hash 14291  df-vtx 29092  df-iedg 29093  df-edg 29142  df-uhgr 29152  df-upgr 29176  df-uspgr 29244  df-nbgr 29427  df-clnbgr 48317  df-isubgr 48359  df-grim 48376  df-gric 48379  df-grlim 48476

This theorem is referenced by:  grlimgredgex  48498