Theorem grilcbri2 48007

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 48000	. . 3 ⊢ (𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ (𝐺 GraphLocIso 𝐻) ≠ ∅)
2		grlimdmrel 47983	. . . . 5 ⊢ Rel dom GraphLocIso
3	2	ovprc 7428	. . . 4 ⊢ (¬ (𝐺 ∈ V ∧ 𝐻 ∈ V) → (𝐺 GraphLocIso 𝐻) = ∅)
4	3	necon1ai 2953	. . 3 ⊢ ((𝐺 GraphLocIso 𝐻) ≠ ∅ → (𝐺 ∈ V ∧ 𝐻 ∈ V))
5	1, 4	sylbi 217	. 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 2730	. . . 4 ⊢ (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑣)
11		eqid 2730	. . . 4 ⊢ (𝐻 ClNeighbVtx (𝑓‘𝑣)) = (𝐻 ClNeighbVtx (𝑓‘𝑣))
12		eqid 2730	. . . 4 ⊢ {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)}
13		eqid 2730	. . . 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 48006	. . 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 2731	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑋 → 𝑗 = 𝑗)
16		oveq2 7398	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑋 → (𝐺 ClNeighbVtx 𝑣) = (𝐺 ClNeighbVtx 𝑋))
17		grilcbri2.n	. . . . . . . . . . . 12 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑋)
18	16, 17	eqtr4di 2783	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑋 → (𝐺 ClNeighbVtx 𝑣) = 𝑁)
19		fveq2 6861	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → (𝑓‘𝑣) = (𝑓‘𝑋))
20	19	oveq2d 7406	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑋 → (𝐻 ClNeighbVtx (𝑓‘𝑣)) = (𝐻 ClNeighbVtx (𝑓‘𝑋)))
21		grilcbri2.m	. . . . . . . . . . . 12 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝑓‘𝑋))
22	20, 21	eqtr4di 2783	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑋 → (𝐻 ClNeighbVtx (𝑓‘𝑣)) = 𝑀)
23	15, 18, 22	f1oeq123d 6797	. . . . . . . . . 10 ⊢ (𝑣 = 𝑋 → (𝑗:(𝐺 ClNeighbVtx 𝑣)–1-1-onto→(𝐻 ClNeighbVtx (𝑓‘𝑣)) ↔ 𝑗:𝑁–1-1-onto→𝑀))
24		eqidd 2731	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → 𝑔 = 𝑔)
25	18	sseq2d 3982	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑋 → ((𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣) ↔ (𝐼‘𝑥) ⊆ 𝑁))
26	25	rabbidv 3416	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ (𝐺 ClNeighbVtx 𝑣)} = 𝐾)
29	22	sseq2d 3982	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑋 → ((𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣)) ↔ (𝐽‘𝑥) ⊆ 𝑀))
30	29	rabbidv 3416	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑋 → {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ (𝐻 ClNeighbVtx (𝑓‘𝑣))} = 𝐿)
33	24, 28, 32	f1oeq123d 6797	. . . . . . . . . . . 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 3301	. . . . . . . . . . . 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 632	. . . . . . . . . . 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 1921	. . . . . . . . . 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 632	. . . . . . . . 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 1921	. . . . . . . 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 3587	. . . . . . 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 612	. . . 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 1917	. . 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 240	. 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 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∅c0 4299   class class class wbr 5110  dom cdm 5641   “ cima 5644  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  iEdgciedg 28931   ClNeighbVtx cclnbgr 47823   GraphLocIso cgrlim 47979   ≃_𝑙𝑔𝑟 cgrlic 47980

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-1o 8437  df-map 8804  df-vtx 28932  df-iedg 28933  df-clnbgr 47824  df-isubgr 47865  df-grim 47882  df-gric 47885  df-grlim 47981  df-grlic 47984

This theorem is referenced by: (None)