Theorem grlimprclnbgr 48238

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 1138	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → {𝐴, 𝐵} ∈ 𝐼)
2		prid1g 4717	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐵})
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → 𝐴 ∈ {𝐴, 𝐵})
4	1, 3	jca 511	. . 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 48237	. . 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 1165	. 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 1195	. . . . 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 1196	. . . . 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 6775	. . . . . . . . . . . 12 ⊢ (𝑓:𝑁–1-1-onto→𝑀 → 𝑓 Fn 𝑁)
16	15	adantl 481	. . . . . . . . . . 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 29257	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
18	17	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
19	18	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐺 ∈ UHGraph)
20	6	eleq2i 2828	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐵} ∈ 𝐼 ↔ {𝐴, 𝐵} ∈ (Edg‘𝐺))
21	20	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ 𝐼 → {𝐴, 𝐵} ∈ (Edg‘𝐺))
22	21	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → {𝐴, 𝐵} ∈ (Edg‘𝐺))
23	22	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → {𝐴, 𝐵} ∈ (Edg‘𝐺))
24	3	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐴 ∈ {𝐴, 𝐵})
25		uhgredgrnv 29203	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ 𝐴 ∈ {𝐴, 𝐵}) → 𝐴 ∈ (Vtx‘𝐺))
26	19, 23, 24, 25	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐴 ∈ (Vtx‘𝐺))
27	26	adantr 480	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
29	28	clnbgrvtxel 48071	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (Vtx‘𝐺) → 𝐴 ∈ (𝐺 ClNeighbVtx 𝐴))
30	27, 29	syl 17	. . . . . . . . . . . 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 2847	. . . . . . . . . . 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 4689	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
33	32	eleq1i 2827	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝐴, 𝐵} ∈ 𝐼 ↔ {𝐵, 𝐴} ∈ 𝐼)
34	33	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ ({𝐴, 𝐵} ∈ 𝐼 → {𝐵, 𝐴} ∈ 𝐼)
35	34	3ad2ant3 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → {𝐵, 𝐴} ∈ 𝐼)
36	35	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → {𝐵, 𝐴} ∈ 𝐼)
37	36	adantr 480	. . . . . . . . . . . . . 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 874	. . . . . . . . . . . . 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 29251	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
40	39	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UPGraph)
41	40	3ad2ant1 1133	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐺 ∈ UPGraph)
42		prid2g 4718	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐴, 𝐵})
43	42	3ad2ant2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼) → 𝐵 ∈ {𝐴, 𝐵})
44	43	3ad2ant3 1135	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐵 ∈ {𝐴, 𝐵})
45		uhgredgrnv 29203	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 ∈ UHGraph ∧ {𝐴, 𝐵} ∈ (Edg‘𝐺) ∧ 𝐵 ∈ {𝐴, 𝐵}) → 𝐵 ∈ (Vtx‘𝐺))
46	19, 23, 44, 45	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ {𝐴, 𝐵} ∈ 𝐼)) → 𝐵 ∈ (Vtx‘𝐺))
47	41, 26, 46	3jca 1128	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 48076	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) → (𝐵 ∈ (𝐺 ClNeighbVtx 𝐴) ↔ (𝐵 = 𝐴 ∨ {𝐵, 𝐴} ∈ 𝐼)))
50	48, 49	syl 17	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 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 2847	. . . . . . . . . . 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 6917	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝑁 ∧ 𝐴 ∈ 𝑁 ∧ 𝐵 ∈ 𝑁) → (𝑓 “ {𝐴, 𝐵}) = {(𝑓‘𝐴), (𝑓‘𝐵)})
54	16, 31, 52, 53	syl3anc 1373	. . . . . . . . . 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 2738	. . . . . . . . 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 229	. . . . . . . 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 25	. . . . . . 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 412	. . . . . 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 1350	. . . . 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 1128	. . . 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 412	. . 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 1920	. 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 15	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {crab 3399   ⊆ wss 3901  {cpr 4582   “ cima 5627   Fn wfn 6487  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  Edgcedg 29120  UHGraphcuhgr 29129  UPGraphcupgr 29153  USPGraphcuspgr 29221   ClNeighbVtx cclnbgr 48060   GraphLocIso cgrlim 48218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9813  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-n0 12402  df-xnn0 12475  df-z 12489  df-uz 12752  df-fz 13424  df-hash 14254  df-vtx 29071  df-iedg 29072  df-edg 29121  df-uhgr 29131  df-upgr 29155  df-uspgr 29223  df-nbgr 29406  df-clnbgr 48061  df-isubgr 48103  df-grim 48120  df-gric 48123  df-grlim 48220

This theorem is referenced by:  grlimprclnbgredg  48239