Theorem grlimprclnbgr 49309

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⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ {(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 4721	. . . . 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 521	. . 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 49308	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ ({𝐴, 𝐵} ∈ 𝐼 ∧ 𝐴 ∈ {𝐴, 𝐵})) → ∃𝑓∃𝑔(𝑓:𝑁⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (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⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (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⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}))) → 𝑓:𝑁⟶1-1onto𝑀)
14		simpr2 1214	. . . . 5 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ (𝑓:𝑁⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}))) → 𝑔:𝐾⟶1-1onto𝐿)
15		f1ofn 6850	. . . . . . . . . . . 12 ⊢ (𝑓:𝑁⟶1-1onto𝑀 → 𝑓 Fn 𝑁)
16	15	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → 𝑓 Fn 𝑁)
17		uspgruhgr 29798	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
18	17	adantr 486	. . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . . . . 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 29741	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ 𝐴 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Vtx‘𝐺))
26	19, 23, 24, 25	syl3anc 1399	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐴 ∈ (Vtx‘𝐺))
27	26	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → 𝐴 ∈ (Vtx‘𝐺))
28		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
29	28	clnbgrvtxel 49142	. . . . . . . . . . . . 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⟶1-1onto𝑀) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
31	30, 5	eleqtrrdi 2875	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → 𝐴 ∈ 𝑁)
32		prcom 4693	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
33	32	eleq1i 2855	. . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → {𝐵, 𝐴} ∈ 𝐼)
38	37	olcd 888	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → (𝐵 = 𝐴 ∨ {𝐵, 𝐴} ∈ 𝐼))
39		uspgrupgr 29792	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
40	39	adantr 486	. . . . . . . . . . . . . . . . 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 4722	. . . . . . . . . . . . . . . . . . 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 29741	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ 𝐵 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Vtx‘𝐺))
46	19, 23, 44, 45	syl3anc 1399	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → (𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)))
49	28, 6	clnbupgrel 49147	. . . . . . . . . . . . . 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⟶1-1onto𝑀) → (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⟶1-1onto𝑀) → 𝐵 ∈ (𝐺 ClNeighbVtx 𝐴))
52	51, 5	eleqtrrdi 2875	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → 𝐵 ∈ 𝑁)
53		fnimapr 6994	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝑁 ∧ 𝐴 ∈ 𝑁 ∧ 𝐵 ∈ 𝑁) → (𝑓 “ {𝐴, 𝐵}) = {(𝑓‘𝐴), (𝑓‘𝐵)})
54	16, 31, 52, 53	syl3anc 1399	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) ∧ 𝑓:𝑁⟶1-1onto𝑀) → (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⟶1-1onto𝑀) → ((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⟶1-1onto𝑀) → ((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⟶1-1onto𝑀) → (𝑔:𝐾⟶1-1onto𝐿 → ((𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}) → {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵}))))
58	57	ex 418	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (𝑓:𝑁⟶1-1onto𝑀 → (𝑔:𝐾⟶1-1onto𝐿 → ((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⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (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⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵}))) → (𝑓:𝑁⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))
61	60	ex 418	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → ((𝑓:𝑁⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵})) → (𝑓:𝑁⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵}))))
62	61	2eximdv 1954	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → (∃𝑓∃𝑔(𝑓:𝑁⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ (𝑓 “ {𝐴, 𝐵}) = (𝑔‘{𝐴, 𝐵})) → ∃𝑓∃𝑔(𝑓:𝑁⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ {(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⟶1-1onto𝑀 ∧ 𝑔:𝐾⟶1-1onto𝐿 ∧ {(𝑓‘𝐴), (𝑓‘𝐵)} = (𝑔‘{𝐴, 𝐵})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {crab 3416   ⊆ wss 3900  {cpr 4590   “ cima 5659   Fn wfn 6556  ⟶1-1ontowf1o 6560  ‘cfv 6561  (class class class)co 7446  Vtxcvtx 29607  Edgcedg 29658  UHGraphcuhgr 29667  UPGraphcupgr 29691  USPGraphcuspgr 29762   ClNeighbVtx cclnbgr 49131   GraphLocIso cgrlim 49289

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 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307

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 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-dju 10012  df-card 10050  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-nn 12366  df-2 12435  df-n0 12637  df-xnn0 12710  df-z 12724  df-uz 12996  df-fz 13670  df-hash 14505  df-vtx 29609  df-iedg 29610  df-edg 29659  df-uhgr 29669  df-upgr 29693  df-uspgr 29764  df-nbgr 29947  df-clnbgr 49132  df-isubgr 49174  df-grim 49191  df-gric 49194  df-grlim 49291

This theorem is used by:  grlimprclnbgredg  49310