Theorem grlimprclnbgr 47981

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 4714	. . . . 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 47980	. . 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 6769	. . . . . . . . . . . 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 29147	. . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . . 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 29093	. . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
29	28	clnbgrvtxel 47814	. . . . . . . . . . . . 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 2839	. . . . . . . . . . 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 4686	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
33	32	eleq1i 2819	. . . . . . . . . . . . . . . . . 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 29141	. . . . . . . . . . . . . . . . . 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 4715	. . . . . . . . . . . . . . . . . . 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 29093	. . . . . . . . . . . . . . . . 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 47819	. . . . . . . . . . . . . 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 2839	. . . . . . . . . . 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 6910	. . . . . . . . . . 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 2731	. . . . . . . . 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 1919	. 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 1540  ∃wex 1779   ∈ wcel 2109  {crab 3396   ⊆ wss 3905  {cpr 4581   “ cima 5626   Fn wfn 6481  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Vtxcvtx 28959  Edgcedg 29010  UHGraphcuhgr 29019  UPGraphcupgr 29043  USPGraphcuspgr 29111   ClNeighbVtx cclnbgr 47803   GraphLocIso cgrlim 47961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-dju 9816  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12754  df-fz 13429  df-hash 14256  df-vtx 28961  df-iedg 28962  df-edg 29011  df-uhgr 29021  df-upgr 29045  df-uspgr 29113  df-nbgr 29296  df-clnbgr 47804  df-isubgr 47846  df-grim 47863  df-gric 47866  df-grlim 47963

This theorem is referenced by:  grlimprclnbgredg  47982