grlimedgclnbgr - Mathbox for Alexander van der Vekens

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Theorem grlimedgclnbgr 48493

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 𝐸 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
grlimedgclnbgr	⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸)))

Distinct variable groups: 𝑥,𝐸 𝑥,𝐼 𝑥,𝑁 𝐴,𝑔,𝑥 𝐴,𝑓 𝑓,𝐸,𝑔 𝑔,𝐹,𝑓 𝑥,𝐹 𝑓,𝐺,𝑔 𝑥,𝐺 𝑓,𝐻,𝑔 𝑥,𝐻 𝑔,𝐼,𝑓 𝑔,𝐽,𝑥 Allowed substitution hints: 𝐽(𝑓) 𝐾(𝑥,𝑓,𝑔) 𝐿(𝑥,𝑓,𝑔) 𝑀(𝑥,𝑓,𝑔) 𝑁(𝑓,𝑔)

Proof of Theorem grlimedgclnbgr Dummy variables 𝑒 𝑣 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1204	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐺 ∈ USPGraph)
2		simp1r 1205	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐻 ∈ USPGraph)
3		simp2 1143	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
4		eqid 2740	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2740	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
6		eqid 2740	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑣)
7		eqid 2740	. . . 4 ⊢ (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐻 ClNeighbVtx (𝐹‘𝑣))
8		clnbgrvtxedg.i	. . . 4 ⊢ 𝐼 = (Edg‘𝐺)
9		grlimedgclnbgr.j	. . . 4 ⊢ 𝐽 = (Edg‘𝐻)
10		sseq1 3947	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ 𝑦 ⊆ (𝐺 ClNeighbVtx 𝑣)))
11	10	cbvrabv 3402	. . . 4 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑦 ∈ 𝐼 ∣ 𝑦 ⊆ (𝐺 ClNeighbVtx 𝑣)}
12		sseq1 3947	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣)) ↔ 𝑦 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))))
13	12	cbvrabv 3402	. . . 4 ⊢ {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} = {𝑦 ∈ 𝐽 ∣ 𝑦 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))}
14	4, 5, 6, 7, 8, 9, 11, 13	usgrlimprop 48491	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
15	1, 2, 3, 14	syl3anc 1379	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → (𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
16		uspgruhgr 29278	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
17	16	adantr 481	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
18	17	3ad2ant1 1139	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐺 ∈ UHGraph)
19	8	eleq2i 2832	. . . . . . . 8 ⊢ (𝐸 ∈ 𝐼 ↔ 𝐸 ∈ (Edg‘𝐺))
20	19	birani 504	. . . . . . 7 ⊢ ((𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ (Edg‘𝐺))
21	20	3ad2ant3 1141	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐸 ∈ (Edg‘𝐺))
22		simp3r 1209	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐴 ∈ 𝐸)
23		uhgredgrnv 29224	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝐴 ∈ 𝐸) → 𝐴 ∈ (Vtx‘𝐺))
24	18, 21, 22, 23	syl3anc 1379	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐴 ∈ (Vtx‘𝐺))
25		eqidd 2741	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → 𝑓 = 𝑓)
26		oveq2 7371	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝐴))
27		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
28	27	oveq2d 7379	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐻 ClNeighbVtx (𝐹‘𝐴)))
29	25, 26, 28	f1oeq123d 6768	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ↔ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))))
30		eqidd 2741	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → 𝑔 = 𝑔)
31	26	sseq2d 3954	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)))
32	31	rabbidv 3399	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
33	28	sseq2d 3954	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣)) ↔ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
34	33	rabbidv 3399	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})
35	30, 32, 34	f1oeq123d 6768	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ↔ 𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}))
36	32	raleqdv 3298	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))
37	35, 36	anbi12d 638	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → ((𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒)) ↔ (𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))))
38	37	exbidv 1928	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒)) ↔ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))))
39	29, 38	anbi12d 638	. . . . . . 7 ⊢ (𝑣 = 𝐴 → ((𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) ↔ (𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
40	39	exbidv 1928	. . . . . 6 ⊢ (𝑣 = 𝐴 → (∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) ↔ ∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
41	40	rspcv 3563	. . . . 5 ⊢ (𝐴 ∈ (Vtx‘𝐺) → (∀𝑣 ∈ (Vtx‘𝐺)∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
42	24, 41	syl 17	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → (∀𝑣 ∈ (Vtx‘𝐺)∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))))
43		eqid 2740	. . . . . . . . . . . . 13 ⊢ 𝑓 = 𝑓
44		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑓 → 𝑓 = 𝑓)
45		clnbgrvtxedg.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝐴)
46	45	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑓 → 𝑁 = (𝐺 ClNeighbVtx 𝐴))
47		grlimedgclnbgr.m	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴))
48	47	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑓 → 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝐴)))
49	44, 46, 48	f1oeq123d 6768	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑓 → (𝑓:𝑁–1-1-onto→𝑀 ↔ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))))
50	43, 49	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑓:𝑁–1-1-onto→𝑀 ↔ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)))
51	50	biimpri 229	. . . . . . . . . . 11 ⊢ (𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) → 𝑓:𝑁–1-1-onto→𝑀)
52	51	adantl 482	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → 𝑓:𝑁–1-1-onto→𝑀)
53	52	adantr 481	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ (𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → 𝑓:𝑁–1-1-onto→𝑀)
54		eqid 2740	. . . . . . . . . . . . 13 ⊢ 𝑔 = 𝑔
55		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑔 → 𝑔 = 𝑔)
56		clnbgrvtxedg.k	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
57	45	sseq2i 3951	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑁 ↔ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴))
58	56, 57	rabbieq 3400	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
59	58	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑔 → 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
60		grlimedgclnbgr.l	. . . . . . . . . . . . . . . 16 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀}
61	47	sseq2i 3951	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑀 ↔ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
62	60, 61	rabbieq 3400	. . . . . . . . . . . . . . 15 ⊢ 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}
63	62	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑔 → 𝐿 = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})
64	55, 59, 63	f1oeq123d 6768	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑔 → (𝑔:𝐾–1-1-onto→𝐿 ↔ 𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))}))
65	54, 64	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑔:𝐾–1-1-onto→𝐿 ↔ 𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})
66	65	biimpri 229	. . . . . . . . . . 11 ⊢ (𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} → 𝑔:𝐾–1-1-onto→𝐿)
67	66	adantr 481	. . . . . . . . . 10 ⊢ ((𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)) → 𝑔:𝐾–1-1-onto→𝐿)
68	67	adantl 482	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ (𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → 𝑔:𝐾–1-1-onto→𝐿)
69		simp3l 1208	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐸 ∈ 𝐼)
70		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ClNeighbVtx 𝐴) = (𝐺 ClNeighbVtx 𝐴)
71		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
72	70, 8, 71	clnbgrvtxedg 48492	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
73	18, 69, 22, 72	syl3anc 1379	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐸 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
74		imaeq2 6015	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝐸 → (𝑓 “ 𝑒) = (𝑓 “ 𝐸))
75		fveq2 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝐸 → (𝑔‘𝑒) = (𝑔‘𝐸))
76	74, 75	eqeq12d 2756	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝐸 → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ 𝐸) = (𝑔‘𝐸)))
77	76	rspcv 3563	. . . . . . . . . . . . 13 ⊢ (𝐸 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} → (∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒) → (𝑓 “ 𝐸) = (𝑔‘𝐸)))
78	73, 77	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → (∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒) → (𝑓 “ 𝐸) = (𝑔‘𝐸)))
79	78	adantld 491	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ((𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)) → (𝑓 “ 𝐸) = (𝑔‘𝐸)))
80	79	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)) → (𝑓 “ 𝐸) = (𝑔‘𝐸)))
81	80	imp 407	. . . . . . . . 9 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ (𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → (𝑓 “ 𝐸) = (𝑔‘𝐸))
82	53, 68, 81	3jca 1134	. . . . . . . 8 ⊢ (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) ∧ (𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸)))
83	82	ex 413	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → ((𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)) → (𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸))))
84	83	eximdv 1924	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) ∧ 𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴))) → (∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒)) → ∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸))))
85	84	expimpd 454	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ((𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸))))
86	85	eximdv 1924	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → (∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸))))
87	42, 86	syld 47	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → (∀𝑣 ∈ (Vtx‘𝐺)∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒))) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸))))
88	87	adantld 491	. 2 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ((𝐹:(Vtx‘𝐺)–1-1-onto→(Vtx‘𝐻) ∧ ∀𝑣 ∈ (Vtx‘𝐺)∃𝑓(𝑓:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} ∧ ∀𝑒 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑓 “ 𝑒) = (𝑔‘𝑒)))) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸))))
89	15, 88	mpd 15	1 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → ∃𝑓∃𝑔(𝑓:𝑁–1-1-onto→𝑀 ∧ 𝑔:𝐾–1-1-onto→𝐿 ∧ (𝑓 “ 𝐸) = (𝑔‘𝐸)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ∀wral 3054 {crab 3392 ⊆ wss 3890 “ cima 5628 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7363 Vtxcvtx 29090 Edgcedg 29141 UHGraphcuhgr 29150 USPGraphcuspgr 29242 ClNeighbVtx cclnbgr 48316 GraphLocIso cgrlim 48474

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 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-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-1o 8402 df-map 8772 df-vtx 29092 df-iedg 29093 df-edg 29142 df-uhgr 29152 df-upgr 29176 df-uspgr 29244 df-clnbgr 48317 df-isubgr 48359 df-grim 48376 df-gric 48379 df-grlim 48476

This theorem is referenced by: grlimprclnbgr 48494

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