Theorem grlimprclnbgrvtx 48475

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 48473	. 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 771	. . . . 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	. . . . . . . . . 10 ⊢ (𝑥 = {(𝑓‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ 𝑀 ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
10	9, 6	elrab2 3637	. . . . . . . . 9 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
11	10	biimpi 216	. . . . . . . 8 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
12	11	adantl 481	. . . . . . 7 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
13	12	adantl 481	. . . . . 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))
14		fvex 6853	. . . . . . . . 9 ⊢ (𝑓‘𝐴) ∈ V
15		fvex 6853	. . . . . . . . 9 ⊢ (𝑓‘𝐵) ∈ V
16	14, 15	prss 4763	. . . . . . . 8 ⊢ (((𝑓‘𝐴) ∈ 𝑀 ∧ (𝑓‘𝐵) ∈ 𝑀) ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀)
17		uspgrupgr 29247	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
18	17	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐻 ∈ UPGraph)
19	18	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐻 ∈ UPGraph)
20	19	ad2antrr 727	. . . . . . . . . . . 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)
21	4	eleq2i 2828	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝐴) ∈ 𝑀 ↔ (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
22	5	clnbupgreli 48311	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ∈ UPGraph ∧ (𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽))
23	22	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽)))
24	21, 23	biimtrid 242	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐴) ∈ 𝑀 → ((𝑓‘𝐴) = (𝐹‘𝐴) ∨ {(𝑓‘𝐴), (𝐹‘𝐴)} ∈ 𝐽)))
25	4	eleq2i 2828	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝐵) ∈ 𝑀 ↔ (𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
26	5	clnbupgreli 48311	. . . . . . . . . . . . . . 15 ⊢ ((𝐻 ∈ UPGraph ∧ (𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽))
27	26	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐵) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)) → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)))
28	25, 27	biimtrid 242	. . . . . . . . . . . . 13 ⊢ (𝐻 ∈ UPGraph → ((𝑓‘𝐵) ∈ 𝑀 → ((𝑓‘𝐵) = (𝐹‘𝐴) ∨ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽)))
29	24, 28	anim12d 610	. . . . . . . . . . . 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))))
30	20, 29	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))))
31	30	imp 406	. . . . . . . . . 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)))
32		prcom 4676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {(𝑓‘𝐴), (𝑓‘𝐵)} = {(𝑓‘𝐵), (𝑓‘𝐴)}
33		preq1 4677	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝑓‘𝐵), (𝑓‘𝐴)} = {(𝐹‘𝐴), (𝑓‘𝐴)})
34	32, 33	eqtrid 2783	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝑓‘𝐴), (𝑓‘𝐵)} = {(𝐹‘𝐴), (𝑓‘𝐴)})
35	34	eleq1d 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝐵) = (𝐹‘𝐴) → ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
36	35	biimpcd 249	. . . . . . . . . . . . . . . . . 18 ⊢ ({(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
37	36	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝑁–1-1-onto→𝑀 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} ∈ 𝐿) → ((𝑓‘𝐵) = (𝐹‘𝐴) → {(𝐹‘𝐴), (𝑓‘𝐴)} ∈ 𝐿))
38	37	adantl 481	. . . . . . . . . . . . . . . 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))
39	38	ad2antrr 727	. . . . . . . . . . . . . . 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))
40		prcom 4676	. . . . . . . . . . . . . . . . . . . 20 ⊢ {(𝑓‘𝐵), (𝐹‘𝐴)} = {(𝐹‘𝐴), (𝑓‘𝐵)}
41	40	eleq1i 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)
42	41	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ ({(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽 → {(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽)
43	42	adantl 481	. . . . . . . . . . . . . . . . 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)
44	20	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . 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)
45		fvex 6853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐹‘𝐴) ∈ V
46	15, 45	pm3.2i 470	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝐵) ∈ V ∧ (𝐹‘𝐴) ∈ V)
47	46	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))
48		simpr 484	. . . . . . . . . . . . . . . . . . . . 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)
49	44, 47, 48	3jca 1129	. . . . . . . . . . . . . . . . . . . 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))
50		eqid 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
51	50, 5	upgrpredgv 29208	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐻 ∈ UPGraph ∧ ((𝑓‘𝐵) ∈ V ∧ (𝐹‘𝐴) ∈ V) ∧ {(𝑓‘𝐵), (𝐹‘𝐴)} ∈ 𝐽) → ((𝑓‘𝐵) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝐴) ∈ (Vtx‘𝐻)))
52		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝐵) ∈ (Vtx‘𝐻) ∧ (𝐹‘𝐴) ∈ (Vtx‘𝐻)) → (𝐹‘𝐴) ∈ (Vtx‘𝐻))
53	49, 51, 52	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))
54	50	clnbgrvtxel 48305	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝐴) ∈ (Vtx‘𝐻) → (𝐹‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
55	4	eleq2i 2828	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝐴) ∈ 𝑀 ↔ (𝐹‘𝐴) ∈ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
56	54, 55	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝐴) ∈ (Vtx‘𝐻) → (𝐹‘𝐴) ∈ 𝑀)
57	53, 56	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)
58		simplrr 778	. . . . . . . . . . . . . . . . . 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)
59	57, 58	prssd 4765	. . . . . . . . . . . . . . . . 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)
60		sseq1 3947	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = {(𝐹‘𝐴), (𝑓‘𝐵)} → (𝑥 ⊆ 𝑀 ↔ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
61	60, 6	elrab2 3637	. . . . . . . . . . . . . . . . 17 ⊢ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐿 ↔ ({(𝐹‘𝐴), (𝑓‘𝐵)} ∈ 𝐽 ∧ {(𝐹‘𝐴), (𝑓‘𝐵)} ⊆ 𝑀))
62	43, 59, 61	sylanbrc 584	. . . . . . . . . . . . . . . 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)
63	62	ex 412	. . . . . . . . . . . . . . 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))
64	39, 63	orim12d 967	. . . . . . . . . . . . . 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)))
65	64	imp 406	. . . . . . . . . . . . 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))
66	65	orcomd 872	. . . . . . . . . . . 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))
67	66	ex 412	. . . . . . . . . . 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)))
68	67	adantld 490	. . . . . . . . . 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)))
69	31, 68	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))
70	69	ex 412	. . . . . . . 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)))
71	16, 70	biimtrrid 243	. . . . . . 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)))
72	71	expimpd 453	. . . . . 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)))
73	13, 72	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))
74	8, 73	jca 511	. . . 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)))
75	74	ex 412	. . 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))))
76	75	eximdv 1919	. 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))))
77	7, 76	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 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {crab 3389  Vcvv 3429   ⊆ wss 3889  {cpr 4569  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  Vtxcvtx 29065  Edgcedg 29116  UPGraphcupgr 29149  USPGraphcuspgr 29217   ClNeighbVtx cclnbgr 48294   GraphLocIso cgrlim 48452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-hash 14293  df-vtx 29067  df-iedg 29068  df-edg 29117  df-uhgr 29127  df-upgr 29151  df-uspgr 29219  df-nbgr 29402  df-clnbgr 48295  df-isubgr 48337  df-grim 48354  df-gric 48357  df-grlim 48454

This theorem is referenced by:  grlimgredgex  48476