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Theorem iscgrg 27763

Description: The congruence property for sequences of points. (Contributed by Thierry Arnoux, 3-Apr-2019.)

**Hypotheses**
Ref	Expression
iscgrg.p	⊢ 𝑃 = (Base‘𝐺)
iscgrg.m	⊢ − = (dist‘𝐺)
iscgrg.e	⊢ ∼ = (cgrG‘𝐺)

**Assertion**
Ref	Expression
iscgrg	⊢ (𝐺 ∈ 𝑉 → (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ (𝑃 ↑_pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))))

Distinct variable groups: 𝑖,𝑗,𝐺 𝐴,𝑖,𝑗 𝐵,𝑖,𝑗 Allowed substitution hints: 𝑃(𝑖,𝑗) ∼ (𝑖,𝑗) − (𝑖,𝑗) 𝑉(𝑖,𝑗)

Proof of Theorem iscgrg Dummy variables 𝑎 𝑏 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscgrg.e	. . . 4 ⊢ ∼ = (cgrG‘𝐺)
2		elex 3493	. . . . 5 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
3		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		iscgrg.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Base‘𝐺)
5	3, 4	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
6	5	oveq1d 7424	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → ((Base‘𝑔) ↑_pm ℝ) = (𝑃 ↑_pm ℝ))
7	6	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑎 ∈ ((Base‘𝑔) ↑_pm ℝ) ↔ 𝑎 ∈ (𝑃 ↑_pm ℝ)))
8	6	eleq2d 2820	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑏 ∈ ((Base‘𝑔) ↑_pm ℝ) ↔ 𝑏 ∈ (𝑃 ↑_pm ℝ)))
9	7, 8	anbi12d 632	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑎 ∈ ((Base‘𝑔) ↑_pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_pm ℝ)) ↔ (𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ))))
10		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
11		iscgrg.m	. . . . . . . . . . . . 13 ⊢ − = (dist‘𝐺)
12	10, 11	eqtr4di 2791	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = − )
13	12	oveqd 7426	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑎‘𝑖) − (𝑎‘𝑗)))
14	12	oveqd 7426	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)))
15	13, 14	eqeq12d 2749	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗)) ↔ ((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))
16	15	2ralbidv 3219	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗)) ↔ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))
17	16	anbi2d 630	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))) ↔ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)))))
18	9, 17	anbi12d 632	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑎 ∈ ((Base‘𝑔) ↑_pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗)))) ↔ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))))
19	18	opabbidv 5215	. . . . . 6 ⊢ (𝑔 = 𝐺 → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑_pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))})
20		df-cgrg 27762	. . . . . 6 ⊢ cgrG = (𝑔 ∈ V ↦ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ ((Base‘𝑔) ↑_pm ℝ) ∧ 𝑏 ∈ ((Base‘𝑔) ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖)(dist‘𝑔)(𝑎‘𝑗)) = ((𝑏‘𝑖)(dist‘𝑔)(𝑏‘𝑗))))})
21		df-xp 5683	. . . . . . . 8 ⊢ ((𝑃 ↑_pm ℝ) × (𝑃 ↑_pm ℝ)) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ))}
22		ovex 7442	. . . . . . . . 9 ⊢ (𝑃 ↑_pm ℝ) ∈ V
23	22, 22	xpex 7740	. . . . . . . 8 ⊢ ((𝑃 ↑_pm ℝ) × (𝑃 ↑_pm ℝ)) ∈ V
24	21, 23	eqeltrri 2831	. . . . . . 7 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ))} ∈ V
25		simpl 484	. . . . . . . 8 ⊢ (((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)))) → (𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)))
26	25	ssopab2i 5551	. . . . . . 7 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))} ⊆ {⟨𝑎, 𝑏⟩ ∣ (𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ))}
27	24, 26	ssexi 5323	. . . . . 6 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))} ∈ V
28	19, 20, 27	fvmpt 6999	. . . . 5 ⊢ (𝐺 ∈ V → (cgrG‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))})
29	2, 28	syl 17	. . . 4 ⊢ (𝐺 ∈ 𝑉 → (cgrG‘𝐺) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))})
30	1, 29	eqtrid 2785	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))})
31	30	breqd 5160	. 2 ⊢ (𝐺 ∈ 𝑉 → (𝐴 ∼ 𝐵 ↔ 𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))}𝐵))
32		dmeq 5904	. . . . . 6 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
33	32	eqeq1d 2735	. . . . 5 ⊢ (𝑎 = 𝐴 → (dom 𝑎 = dom 𝑏 ↔ dom 𝐴 = dom 𝑏))
34	32	adantr 482	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 ∈ dom 𝑎) → dom 𝑎 = dom 𝐴)
35		simpll 766	. . . . . . . . . 10 ⊢ (((𝑎 = 𝐴 ∧ 𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → 𝑎 = 𝐴)
36	35	fveq1d 6894	. . . . . . . . 9 ⊢ (((𝑎 = 𝐴 ∧ 𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (𝑎‘𝑖) = (𝐴‘𝑖))
37	35	fveq1d 6894	. . . . . . . . 9 ⊢ (((𝑎 = 𝐴 ∧ 𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (𝑎‘𝑗) = (𝐴‘𝑗))
38	36, 37	oveq12d 7427	. . . . . . . 8 ⊢ (((𝑎 = 𝐴 ∧ 𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → ((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝐴‘𝑖) − (𝐴‘𝑗)))
39	38	eqeq1d 2735	. . . . . . 7 ⊢ (((𝑎 = 𝐴 ∧ 𝑖 ∈ dom 𝑎) ∧ 𝑗 ∈ dom 𝑎) → (((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)) ↔ ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))
40	34, 39	raleqbidva 3328	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 ∈ dom 𝑎) → (∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)) ↔ ∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))
41	32, 40	raleqbidva 3328	. . . . 5 ⊢ (𝑎 = 𝐴 → (∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)) ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))
42	33, 41	anbi12d 632	. . . 4 ⊢ (𝑎 = 𝐴 → ((dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))) ↔ (dom 𝐴 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)))))
43		dmeq 5904	. . . . . 6 ⊢ (𝑏 = 𝐵 → dom 𝑏 = dom 𝐵)
44	43	eqeq2d 2744	. . . . 5 ⊢ (𝑏 = 𝐵 → (dom 𝐴 = dom 𝑏 ↔ dom 𝐴 = dom 𝐵))
45		fveq1 6891	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏‘𝑖) = (𝐵‘𝑖))
46		fveq1 6891	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏‘𝑗) = (𝐵‘𝑗))
47	45, 46	oveq12d 7427	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝑏‘𝑖) − (𝑏‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))
48	47	eqeq2d 2744	. . . . . 6 ⊢ (𝑏 = 𝐵 → (((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)) ↔ ((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
49	48	2ralbidv 3219	. . . . 5 ⊢ (𝑏 = 𝐵 → (∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗)) ↔ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))
50	44, 49	anbi12d 632	. . . 4 ⊢ (𝑏 = 𝐵 → ((dom 𝐴 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
51	42, 50	sylan9bb 511	. . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))) ↔ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
52		eqid 2733	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))}
53	51, 52	brab2a 5770	. 2 ⊢ (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ↑_pm ℝ) ∧ 𝑏 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝑎 = dom 𝑏 ∧ ∀𝑖 ∈ dom 𝑎∀𝑗 ∈ dom 𝑎((𝑎‘𝑖) − (𝑎‘𝑗)) = ((𝑏‘𝑖) − (𝑏‘𝑗))))}𝐵 ↔ ((𝐴 ∈ (𝑃 ↑_pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗)))))
54	31, 53	bitrdi 287	1 ⊢ (𝐺 ∈ 𝑉 → (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ (𝑃 ↑_pm ℝ) ∧ 𝐵 ∈ (𝑃 ↑_pm ℝ)) ∧ (dom 𝐴 = dom 𝐵 ∧ ∀𝑖 ∈ dom 𝐴∀𝑗 ∈ dom 𝐴((𝐴‘𝑖) − (𝐴‘𝑗)) = ((𝐵‘𝑖) − (𝐵‘𝑗))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 class class class wbr 5149 {copab 5211 × cxp 5675 dom cdm 5677 ‘cfv 6544 (class class class)co 7409 ↑_pm cpm 8821 ℝcr 11109 Basecbs 17144 distcds 17206 cgrGccgrg 27761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-cgrg 27762

This theorem is referenced by: iscgrgd 27764 ercgrg 27768

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