grlimedgclnbgr - Mathbox for Alexander van der Vekens

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Theorem grlimedgclnbgr 48801

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 1216	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐺 ∈ USPGraph)
2		simp1r 1217	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐻 ∈ USPGraph)
3		simp2 1155	. . 3 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻))
4		eqid 2766	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2766	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
6		eqid 2766	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑣)
7		eqid 2766	. . . 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 3965	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ 𝑦 ⊆ (𝐺 ClNeighbVtx 𝑣)))
11	10	cbvrabv 3429	. . . 4 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑦 ∈ 𝐼 ∣ 𝑦 ⊆ (𝐺 ClNeighbVtx 𝑣)}
12		sseq1 3965	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣)) ↔ 𝑦 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))))
13	12	cbvrabv 3429	. . . 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 48799	. . 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 1398	. 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 29571	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
17	16	adantr 486	. . . . . . 7 ⊢ ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
18	17	3ad2ant1 1151	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐺 ∈ UHGraph)
19	8	eleq2i 2858	. . . . . . . 8 ⊢ (𝐸 ∈ 𝐼 ↔ 𝐸 ∈ (Edg‘𝐺))
20	19	birani 509	. . . . . . 7 ⊢ ((𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ (Edg‘𝐺))
21	20	3ad2ant3 1153	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐸 ∈ (Edg‘𝐺))
22		simp3r 1221	. . . . . 6 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐴 ∈ 𝐸)
23		uhgredgrnv 29517	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝐴 ∈ 𝐸) → 𝐴 ∈ (Vtx‘𝐺))
24	18, 21, 22, 23	syl3anc 1398	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐴 ∈ (Vtx‘𝐺))
25		eqidd 2767	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → 𝑓 = 𝑓)
26		oveq2 7431	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝐴))
27		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
28	27	oveq2d 7439	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐻 ClNeighbVtx (𝐹‘𝐴)))
29	25, 26, 28	f1oeq123d 6821	. . . . . . . 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 2767	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → 𝑔 = 𝑔)
31	26	sseq2d 3972	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)))
32	31	rabbidv 3426	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
33	28	sseq2d 3972	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣)) ↔ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
34	33	rabbidv 3426	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})
35	30, 32, 34	f1oeq123d 6821	. . . . . . . . . 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 3326	. . . . . . . . . 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 644	. . . . . . . . 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 1954	. . . . . . . 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 644	. . . . . . 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 1954	. . . . . 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 3580	. . . . 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 18	. . . 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 2766	. . . . . . . . . . . . 13 ⊢ 𝑓 = 𝑓
44		id 23	. . . . . . . . . . . . . 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 6821	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (𝑓:(𝐺 ClNeighbVtx 𝐴)–1-1-onto→(𝐻 ClNeighbVtx (𝐹‘𝐴)) → 𝑓:𝑁–1-1-onto→𝑀)
52	51	adantl 487	. . . . . . . . . 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 486	. . . . . . . . 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 2766	. . . . . . . . . . . . 13 ⊢ 𝑔 = 𝑔
55		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑔 → 𝑔 = 𝑔)
56		clnbgrvtxedg.k	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
57	45	sseq2i 3969	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑁 ↔ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴))
58	56, 57	rabbieq 3427	. . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑀 ↔ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
62	60, 61	rabbieq 3427	. . . . . . . . . . . . . . 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 6821	. . . . . . . . . . . . 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 231	. . . . . . . . . . 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 486	. . . . . . . . . 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 487	. . . . . . . . 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 1220	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐸 ∈ 𝐼)
70		eqid 2766	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ClNeighbVtx 𝐴) = (𝐺 ClNeighbVtx 𝐴)
71		eqid 2766	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
72	70, 8, 71	clnbgrvtxedg 48800	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸) → 𝐸 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
73	18, 69, 22, 72	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻) ∧ (𝐸 ∈ 𝐼 ∧ 𝐴 ∈ 𝐸)) → 𝐸 ∈ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
74		imaeq2 6063	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝐸 → (𝑓 “ 𝑒) = (𝑓 “ 𝐸))
75		fveq2 6888	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝐸 → (𝑔‘𝑒) = (𝑔‘𝐸))
76	74, 75	eqeq12d 2782	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝐸 → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ 𝐸) = (𝑔‘𝐸)))
77	76	rspcv 3580	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 496	. . . . . . . . . . 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 486	. . . . . . . . . 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 412	. . . . . . . . 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 1146	. . . . . . . 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 418	. . . . . . 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 1950	. . . . . 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 459	. . . . 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 1950	. . . 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 48	. . 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 496	. 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 16	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ∀wral 3082 {crab 3419 ⊆ wss 3908 “ cima 5669 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7423 Vtxcvtx 29383 Edgcedg 29434 UHGraphcuhgr 29443 USPGraphcuspgr 29535 ClNeighbVtx cclnbgr 48624 GraphLocIso cgrlim 48782

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 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7426 df-oprab 7427 df-mpo 7428 df-1st 7995 df-2nd 7996 df-1o 8462 df-map 8835 df-vtx 29385 df-iedg 29386 df-edg 29435 df-uhgr 29445 df-upgr 29469 df-uspgr 29537 df-clnbgr 48625 df-isubgr 48667 df-grim 48684 df-gric 48687 df-grlim 48784

This theorem is used by: grlimprclnbgr 48802

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