Theorem grlimprclnbgr 48738

Description: For two locally isomorphic graphs 𝐺 and 𝐻 and a vertex 𝐴 of 𝐺 there are two bijections 𝑓 and 𝑔 mapping the closed neighborhood 𝑁 of 𝐴 onto the closed neighborhood 𝑀 of (𝐹‘𝐴) and the edges between the vertices in 𝑁 onto the edges between the vertices in 𝑀, so that the mapped vertices of an edge {𝐴, 𝐵} containing the vertex 𝐴 is an edge between the vertices in 𝑀. (Contributed by AV, 25-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
grlimprclnbgr	⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))

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

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

Proof of Theorem grlimprclnbgr

Step	Hyp	Ref	Expression
1		simp3 1156	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → {𝐴, 𝐵} ∈ 𝐼)
2		prid1g 4727	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
3	2	3ad2ant1 1151	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → 𝐴 ∈ {𝐴, 𝐵})
4	1, 3	jca 520	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → ({𝐴, 𝐵} ∈ 𝐼 ∧ 𝐴 ∈ {𝐴, 𝐵}))
5		clnbgrvtxedg.n	. . . 4 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
6		clnbgrvtxedg.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐺)
7		clnbgrvtxedg.k	. . . 4 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
8		grlimedgclnbgr.m	. . . 4 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))
9		grlimedgclnbgr.j	. . . 4 ⊢ 𝐽 = (Edg‘𝐻)
10		grlimedgclnbgr.l	. . . 4 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
11	5, 6, 7, 8, 9, 10	grlimedgclnbgr 48737	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ ({𝐴, 𝐵} ∈ 𝐼 ∧ 𝐴 ∈ {𝐴, 𝐵})) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵})))
12	4, 11	syl3an3 1183	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵})))
13		simpr1 1213	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}))) → 𝑓:𝑁–1-1-onto→𝑀)
14		simpr2 1214	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}))) → 𝑔:𝐾–1-1-onto→𝐿)
15		f1ofn 6823	. . . . . . . . . . . 12 ⊢ (𝑓:𝑁–1-1-onto→𝑀 → 𝑓 Fn 𝑁)
16	15	adantl 486	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → 𝑓 Fn 𝑁)
17		uspgruhgr 29512	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
18	17	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
19	18	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐺 ∈ UHGraph)
20	6	eleq2i 2855	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐵} ∈ 𝐼 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
21	20	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ 𝐼 → {𝐴, 𝐵} ∈ (Edg‘𝐺))
22	21	3ad2ant3 1153	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → {𝐴, 𝐵} ∈ (Edg‘𝐺))
23	22	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → {𝐴, 𝐵} ∈ (Edg‘𝐺))
24	3	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐴 ∈ {𝐴, 𝐵})
25		uhgredgrnv 29458	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ 𝐴 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Vtx‘𝐺))
26	19, 23, 24, 25	syl3anc 1398	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐴 ∈ (Vtx‘𝐺))
27	26	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → 𝐴 ∈ (Vtx‘𝐺))
28		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
29	28	clnbgrvtxel 48571	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Vtx‘𝐺) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
30	27, 29	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
31	30, 5	eleqtrrdi 2874	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → 𝐴 ∈ 𝑁)
32		prcom 4699	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
33	32	eleq1i 2854	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐵} ∈ 𝐼 ↔ {𝐵, 𝐴} ∈ 𝐼)
34	33	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ 𝐼 → {𝐵, 𝐴} ∈ 𝐼)
35	34	3ad2ant3 1153	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → {𝐵, 𝐴} ∈ 𝐼)
36	35	3ad2ant3 1153	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → {𝐵, 𝐴} ∈ 𝐼)
37	36	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → {𝐵, 𝐴} ∈ 𝐼)
38	37	olcd 887	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → (𝐵 = 𝐴 ∨ {𝐵, 𝐴} ∈ 𝐼))
39		uspgrupgr 29506	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
40	39	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UPGraph)
41	40	3ad2ant1 1151	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐺 ∈ UPGraph)
42		prid2g 4728	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
43	42	3ad2ant2 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → 𝐵 ∈ {𝐴, 𝐵})
44	43	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐵 ∈ {𝐴, 𝐵})
45		uhgredgrnv 29458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ 𝐵 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Vtx‘𝐺))
46	19, 23, 44, 45	syl3anc 1398	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐵 ∈ (Vtx‘𝐺))
47	41, 26, 46	3jca 1146	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
48	47	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
49	28, 6	clnbupgrel 48576	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐵 ∈ (𝐺 ClNeighbVtx 𝐴) ↔ (𝐵 = 𝐴 ∨ {𝐵, 𝐴} ∈ 𝐼)))
50	48, 49	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → (𝐵 ∈ (𝐺 ClNeighbVtx 𝐴) ↔ (𝐵 = 𝐴 ∨ {𝐵, 𝐴} ∈ 𝐼)))
51	38, 50	mpbird 260	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
52	51, 5	eleqtrrdi 2874	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → 𝐵 ∈ 𝑁)
53		fnimapr 6966	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝑁 ∧ 𝐴 ∈ 𝑁 ∧ 𝐵 ∈ 𝑁) → (𝑓 “ {𝐴, 𝐵}) = {(𝑓‘𝐴), (𝑓‘𝐵)})
54	16, 31, 52, 53	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → (𝑓 “ {𝐴, 𝐵}) = {(𝑓‘𝐴), (𝑓‘𝐵)})
55	54	eqeq1d 2765	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → ((𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}) ↔ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))
56	55	biimpd 232	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → ((𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}) → {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))
57	56	a1d 26	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁–1-1-onto→𝑀) → (𝑔:𝐾–1-1-onto→𝐿 → ((𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}) → {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵}))))
58	57	ex 417	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (𝑓:𝑁–1-1-onto→𝑀 → (𝑔:𝐾–1-1-onto→𝐿 → ((𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}) → {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))))
59	58	3imp2 1368	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}))) → {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵}))
60	13, 14, 59	3jca 1146	. . . 4 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}))) → (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))
61	60	ex 417	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ((𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵})) → (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵}))))
62	61	2eximdv 1949	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵})) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵}))))
63	12, 62	mpd 16	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {crab 3416   ⊆ wss 3906  {cpr 4592   “ cima 5666   Fn wfn 6533  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  Vtxcvtx 29324  Edgcedg 29375  UHGraphcuhgr 29384  UPGraphcupgr 29408  USPGraphcuspgr 29476   ClNeighbVtx cclnbgr 48560   GraphLocIso cgrlim 48718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-dju 9888  df-card 9926  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-2 12304  df-n0 12506  df-xnn0 12579  df-z 12593  df-uz 12864  df-fz 13537  df-hash 14369  df-vtx 29326  df-iedg 29327  df-edg 29376  df-uhgr 29386  df-upgr 29410  df-uspgr 29478  df-nbgr 29661  df-clnbgr 48561  df-isubgr 48603  df-grim 48620  df-gric 48623  df-grlim 48720

This theorem is referenced by:  grlimprclnbgredg  48739