Theorem grlimprclnbgrvtx 49302

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⟶1-1onto𝑀 ∧ ({(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 49300	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿))
8		simprl 783	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → 𝑓:𝑁⟶1-1onto𝑀)
9		sseq1 3938	. . . . . . . . 9 ⊢ (𝑥 = {(𝑓‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ 𝑀 ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
10	9, 6	elrab2 3652	. . . . . . . 8 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
11	10	bilani 510	. . . . . . 7 ⊢ ((𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
12	11	adantl 487	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
13		fvex 6921	. . . . . . . . 9 ⊢ (𝑓‘𝐴) ∈ V
14		fvex 6921	. . . . . . . . 9 ⊢ (𝑓‘𝐵) ∈ V
15	13, 14	prss 4779	. . . . . . . 8 ⊢ (((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀) ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀)
16		uspgrupgr 29789	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
17	16	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐻 ∈ UPGraph)
18	17	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐻 ∈ UPGraph)
19	18	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽) → 𝐻 ∈ UPGraph)
20	4	eleq2i 2856	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝐴) ∈ 𝑀 ↔ (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
21	5	clnbupgreli 49138	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ∈ UPGraph ∧ (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽))
22	21	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽)))
23	20, 22	biimtrid 245	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐴) ∈ 𝑀 → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽)))
24	4	eleq2i 2856	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝐵) ∈ 𝑀 ↔ (𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
25	5	clnbupgreli 49138	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ∈ UPGraph ∧ (𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽))
26	25	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)))
27	24, 26	biimtrid 245	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐵) ∈ 𝑀 → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)))
28	23, 27	anim12d 621	. . . . . . . . . . . 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 18	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 412	. . . . . . . . . 10 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 4693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {(𝑓‘𝐴), (𝑓‘𝐵)} = {(𝑓‘𝐵), (𝑓‘𝐴)}
32		preq1 4694	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝑓‘𝐵), (𝑓‘𝐴)} = {(𝐹‘𝐴), (𝑓‘𝐴)})
33	31, 32	eqtrid 2810	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝑓‘𝐴), (𝑓‘𝐵)} = {(𝐹‘𝐴), (𝑓‘𝐴)})
34	33	eleq1d 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
35	34	biimpcd 252	. . . . . . . . . . . . . . . . . 18 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
36	35	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
37	36	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
38	37	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 4693	. . . . . . . . . . . . . . . . . . 19 ⊢ {(𝑓‘𝐵), (𝐹‘𝐴)} = {(𝐹‘𝐴), (𝑓‘𝐵)}
40	39	eleq1i 2855	. . . . . . . . . . . . . . . . . 18 ⊢ ({(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)
41	40	bilani 510	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 6921	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹‘𝐴) ∈ V
44	14, 43	pm3.2i 476	. . . . . . . . . . . . . . . . . . . . . 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⟶1-1onto𝑀 ∧ {(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 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 1146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 2763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
49	48, 5	upgrpredgv 29747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 ∈ UPGraph ∧ ((𝑓‘𝐵) ∈ V ∧ (𝐹‘𝐴) ∈ V) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → ((𝑓‘𝐵) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝐴) ∈ (Vtx‘𝐻)))
50		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝐵) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝐴) ∈ (Vtx‘𝐻)) → (𝐹‘𝐴) ∈ (Vtx‘𝐻))
51	47, 49, 50	3syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 49132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝐴) ∈ (Vtx‘𝐻) → (𝐹‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
53	4	eleq2i 2856	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝐴) ∈ 𝑀 ↔ (𝐹‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
54	52, 53	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝐴) ∈ (Vtx‘𝐻) → (𝐹‘𝐴) ∈ 𝑀)
55	51, 54	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 790	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 4781	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 3938	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ 𝑀 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
59	58, 6	elrab2 3652	. . . . . . . . . . . . . . . . 17 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
60	41, 57, 59	sylanbrc 595	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 418	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 979	. . . . . . . . . . . . . 14 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 412	. . . . . . . . . . . . 13 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 885	. . . . . . . . . . . 12 ⊢ (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 418	. . . . . . . . . . 11 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 496	. . . . . . . . . 10 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 16	. . . . . . . . 9 ⊢ ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 418	. . . . . . . 8 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 246	. . . . . . 7 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 459	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(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 16	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
72	8, 71	jca 521	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿)) → (𝑓:𝑁⟶1-1onto𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))
73	72	ex 418	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ((𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → (𝑓:𝑁⟶1-1onto𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))))
74	73	eximdv 1952	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (∃𝑓(𝑓:𝑁⟶1-1onto𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ∃𝑓(𝑓:𝑁⟶1-1onto𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))))
75	7, 74	mpd 16	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓(𝑓:𝑁⟶1-1onto𝑀 ∧ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ∨ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900  {cpr 4590  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443  Vtxcvtx 29604  Edgcedg 29655  UPGraphcupgr 29688  USPGraphcuspgr 29759   ClNeighbVtx cclnbgr 49121   GraphLocIso cgrlim 49279

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-fz 13667  df-hash 14502  df-vtx 29606  df-iedg 29607  df-edg 29656  df-uhgr 29666  df-upgr 29690  df-uspgr 29761  df-nbgr 29944  df-clnbgr 49122  df-isubgr 49164  df-grim 49181  df-gric 49184  df-grlim 49281

This theorem is used by:  grlimgredgex  49303