Theorem grilcbri2 48581

Description: Implications of two graphs being locally isomorphic. (Contributed by AV, 9-Jun-2025.)

Hypotheses

Ref	Expression
dfgrlic2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfgrlic2.w	⊢ 𝑊 = (Vtx‘𝐻)
dfgrlic3.i	⊢ 𝐼 = (iEdg‘𝐺)
dfgrlic3.j	⊢ 𝐽 = (iEdg‘𝐻)
grilcbri2.n	⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)
grilcbri2.m	⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑋))
grilcbri2.k	⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
grilcbri2.l	⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}

Assertion

Ref	Expression
grilcbri2	⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))

Distinct variable groups:   𝑓,𝐺   𝑓,𝐻   𝑓,𝑋   𝑔,𝐺,𝑖,𝑗,𝑓   𝑥,𝐺   𝑔,𝐻,𝑖,𝑗   𝑥,𝐻   𝑖,𝐼,𝑥   𝑖,𝐽,𝑥   𝑖,𝐾   𝑔,𝑋,𝑗   𝑥,𝑋,𝑓

Allowed substitution hints:   𝐼(𝑓,𝑔,𝑗)   𝐽(𝑓,𝑔,𝑗)   𝐾(𝑥,𝑓,𝑔,𝑗)   𝐿(𝑥,𝑓,𝑔,𝑖,𝑗)   𝑀(𝑥,𝑓,𝑔,𝑖,𝑗)   𝑁(𝑥,𝑓,𝑔,𝑖,𝑗)   𝑉(𝑥,𝑓,𝑔,𝑖,𝑗)   𝑊(𝑥,𝑓,𝑔,𝑖,𝑗)   𝑋(𝑖)

Proof of Theorem grilcbri2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brgrlic 48574	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2		grlimdmrel 48550	. . . . 5 ⊢ Rel dom GraphLocIso
3	2	ovprc 7423	. . . 4 ⊢ (¬ (𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺 GraphLocIso 𝐻) = ∅)
4	3	necon1ai 2978	. . 3 ⊢ ((𝐺 GraphLocIso 𝐻) ≠ ∅ → (𝐺 ∈ V ∧ 𝐻 ∈ V))
5	1, 4	sylbi 219	. 2 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → (𝐺 ∈ V ∧ 𝐻 ∈ V))
6		dfgrlic2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
7		dfgrlic2.w	. . . 4 ⊢ 𝑊 = (Vtx‘𝐻)
8		dfgrlic3.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
9		dfgrlic3.j	. . . 4 ⊢ 𝐽 = (iEdg‘𝐻)
10		eqid 2756	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑣)
11		eqid 2756	. . . 4 ⊢ (𝐻 ClNeighbVtx (𝑓‘𝑣)) = (𝐻 ClNeighbVtx (𝑓‘𝑣))
12		eqid 2756	. . . 4 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}
13		eqid 2756	. . . 4 ⊢ {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))}
14	6, 7, 8, 9, 10, 11, 12, 13	dfgrlic3 48580	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
15		eqidd 2757	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑋 → 𝑗 = 𝑗)
16		oveq2 7393	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑋 → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑋))
17		grilcbri2.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)
18	16, 17	eqtr4di 2809	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑋 → (𝐺 ClNeighbVtx 𝑣) = 𝑁)
19		fveq2 6856	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → (𝑓‘𝑣) = (𝑓‘𝑋))
20	19	oveq2d 7401	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑋 → (𝐻 ClNeighbVtx (𝑓‘𝑣)) = (𝐻 ClNeighbVtx (𝑓‘𝑋)))
21		grilcbri2.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑋))
22	20, 21	eqtr4di 2809	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑋 → (𝐻 ClNeighbVtx (𝑓‘𝑣)) = 𝑀)
23	15, 18, 22	f1oeq123d 6789	. . . . . . . . . 10 ⊢ (𝑣 = 𝑋 → (𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ↔ 𝑗:𝑁–1-1-onto→𝑀))
24		eqidd 2757	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → 𝑔 = 𝑔)
25	18	sseq2d 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑋 → ((𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ (𝐼‘𝑥) ⊆ 𝑁))
26	25	rabbidv 3415	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑋 → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁})
27		grilcbri2.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁}
28	26, 27	eqtr4di 2809	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} = 𝐾)
29	22	sseq2d 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑋 → ((𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣)) ↔ (𝐽‘𝑥) ⊆ 𝑀))
30	29	rabbidv 3415	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑋 → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀})
31		grilcbri2.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀}
32	30, 31	eqtr4di 2809	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} = 𝐿)
33	24, 28, 32	f1oeq123d 6789	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑋 → (𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ↔ 𝑔:𝐾–1-1-onto→𝐿))
34	28	raleqdv 3314	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑋 → (∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)) ↔ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))
35	33, 34	anbi12d 640	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑋 → ((𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))) ↔ (𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
36	35	exbidv 1935	. . . . . . . . . 10 ⊢ (𝑣 = 𝑋 → (∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))) ↔ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))
37	23, 36	anbi12d 640	. . . . . . . . 9 ⊢ (𝑣 = 𝑋 → ((𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))) ↔ (𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
38	37	exbidv 1935	. . . . . . . 8 ⊢ (𝑣 = 𝑋 → (∃𝑗(𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))) ↔ ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
39	38	rspcv 3572	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))) → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
40	39	com12 32	. . . . . 6 ⊢ (∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))) → (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))
41	40	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))) → (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))
42	41	anim2d 620	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → ((𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))) → (𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))))
43	42	eximdv 1931	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑗(𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ∧ ∃𝑔(𝑔:{𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}–1-1-onto→{𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} ∧ ∀𝑖 ∈ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))) → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))))
44	14, 43	sylbid 242	. 2 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))))
45	5, 44	mpcom 38	1 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 → ∃𝑓(𝑓:𝑉–1-1-onto→𝑊 ∧ (𝑋 ∈ 𝑉 → ∃𝑗(𝑗:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑗 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554  ∃wex 1793   ∈ wcel 2136   ≠ wne 2951  ∀wral 3070  {crab 3408  Vcvv 3448   ⊆ wss 3899  ∅c0 4280   class class class wbr 5094  dom cdm 5640   “ cima 5643  –1-1-onto→wf1o 6509  ‘cfv 6510  (class class class)co 7385  Vtxcvtx 29136  iEdgciedg 29137   ClNeighbVtx cclnbgr 48388   GraphLocIso cgrlim 48546   ≃_𝑙𝑔𝑟 cgrlic 48547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-1o 8425  df-map 8798  df-vtx 29138  df-iedg 29139  df-clnbgr 48389  df-isubgr 48431  df-grim 48448  df-gric 48451  df-grlim 48548  df-grlic 48551

This theorem is referenced by: (None)