grlimedgclnbgr - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > grlimedgclnbgr		Structured version Visualization version GIF version

Theorem grlimedgclnbgr 48737

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 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5		eqid 2763	. . . 4 ⊢ (Vtx‘𝐻) = (Vtx‘𝐻)
6		eqid 2763	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑣)
7		eqid 2763	. . . 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 3963	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ 𝑦 ⊆ (𝐺 ClNeighbVtx 𝑣)))
11	10	cbvrabv 3426	. . . 4 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑦 ∈ 𝐼 ∣ 𝑦 ⊆ (𝐺 ClNeighbVtx 𝑣)}
12		sseq1 3963	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣)) ↔ 𝑦 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))))
13	12	cbvrabv 3426	. . . 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 48735	. . 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 29512	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
17	16	adantr 485	. . . . . . 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 2855	. . . . . . . 8 ⊢ (𝐸 ∈ 𝐼 ↔ 𝐸 ∈ (Edg‘𝐺))
20	19	birani 508	. . . . . . 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 29458	. . . . . 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 2764	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → 𝑓 = 𝑓)
26		oveq2 7420	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝐴))
27		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
28	27	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → (𝐻 ClNeighbVtx (𝐹‘𝑣)) = (𝐻 ClNeighbVtx (𝐹‘𝐴)))
29	25, 26, 28	f1oeq123d 6816	. . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → 𝑔 = 𝑔)
31	26	sseq2d 3970	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)))
32	31	rabbidv 3423	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)})
33	28	sseq2d 3970	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝐴 → (𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣)) ↔ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))))
34	33	rabbidv 3423	. . . . . . . . . . 11 ⊢ (𝑣 = 𝐴 → {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝑣))} = {𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴))})
35	30, 32, 34	f1oeq123d 6816	. . . . . . . . . 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 3323	. . . . . . . . . 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 643	. . . . . . . . 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 1951	. . . . . . . 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 643	. . . . . . 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 1951	. . . . . 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 3578	. . . . 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 2763	. . . . . . . . . . . . 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 6816	. . . . . . . . . . . . 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 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))) → 𝑓:𝑁–1-1-onto→𝑀)
53	52	adantr 485	. . . . . . . . 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 2763	. . . . . . . . . . . . 13 ⊢ 𝑔 = 𝑔
55		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑔 → 𝑔 = 𝑔)
56		clnbgrvtxedg.k	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁}
57	45	sseq2i 3967	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑁 ↔ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴))
58	56, 57	rabbieq 3424	. . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑀 ↔ 𝑥 ⊆ (𝐻 ClNeighbVtx (𝐹‘𝐴)))
62	60, 61	rabbieq 3424	. . . . . . . . . . . . . . 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 6816	. . . . . . . . . . . . 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 485	. . . . . . . . . 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 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))) → 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 2763	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ClNeighbVtx 𝐴) = (𝐺 ClNeighbVtx 𝐴)
71		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)} = {𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ (𝐺 ClNeighbVtx 𝐴)}
72	70, 8, 71	clnbgrvtxedg 48736	. . . . . . . . . . . . . 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 6060	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝐸 → (𝑓 “ 𝑒) = (𝑓 “ 𝐸))
75		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝐸 → (𝑔‘𝑒) = (𝑔‘𝐸))
76	74, 75	eqeq12d 2779	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝐸 → ((𝑓 “ 𝑒) = (𝑔‘𝑒) ↔ (𝑓 “ 𝐸) = (𝑔‘𝐸)))
77	76	rspcv 3578	. . . . . . . . . . . . 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 495	. . . . . . . . . . 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 485	. . . . . . . . . 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 411	. . . . . . . . 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 417	. . . . . . 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 1947	. . . . . 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 458	. . . . 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 1947	. . . 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 495	. 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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ∀wral 3079 {crab 3416 ⊆ wss 3906 “ cima 5666 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 Vtxcvtx 29324 Edgcedg 29375 UHGraphcuhgr 29384 USPGraphcuspgr 29476 ClNeighbVtx cclnbgr 48560 GraphLocIso cgrlim 48718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-1o 8454 df-map 8827 df-vtx 29326 df-iedg 29327 df-edg 29376 df-uhgr 29386 df-upgr 29410 df-uspgr 29478 df-clnbgr 48561 df-isubgr 48603 df-grim 48620 df-gric 48623 df-grlim 48720

This theorem is referenced by: grlimprclnbgr 48738

W3C validator