Theorem hbtlem1 36712

Description: Value of the leading coefficient sequence function. (Contributed by Stefan O'Rear, 31-Mar-2015.)

Hypotheses

Ref	Expression
hbtlem.p	⊢ 𝑃 = (Poly1‘𝑅)
hbtlem.u	⊢ 𝑈 = (LIdeal‘𝑃)
hbtlem.s	⊢ 𝑆 = (ldgIdlSeq‘𝑅)
hbtlem.d	⊢ 𝐷 = ( deg1 ‘𝑅)

Assertion

Ref	Expression
hbtlem1	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})

Distinct variable groups:   𝑗,𝐼,𝑘   𝑅,𝑗,𝑘   𝑗,𝑋,𝑘

Allowed substitution hints:   𝐷(𝑗,𝑘)   𝑃(𝑗,𝑘)   𝑆(𝑗,𝑘)   𝑈(𝑗,𝑘)   𝑉(𝑗,𝑘)

Proof of Theorem hbtlem1

Dummy variables 𝑖 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbtlem.s	. . . . . 6 ⊢ 𝑆 = (ldgIdlSeq‘𝑅)
2		elex 3185	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
3		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
4		hbtlem.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (Poly1‘𝑅)
5	3, 4	syl6eqr 2662	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
6	5	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = (LIdeal‘𝑃))
7		hbtlem.u	. . . . . . . . . 10 ⊢ 𝑈 = (LIdeal‘𝑃)
8	6, 7	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (LIdeal‘(Poly1‘𝑟)) = 𝑈)
9		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
10		hbtlem.d	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = ( deg1 ‘𝑅)
11	9, 10	syl6eqr 2662	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = 𝐷)
12	11	fveq1d 6105	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑅 → (( deg1 ‘𝑟)‘𝑘) = (𝐷‘𝑘))
13	12	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑅 → ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑥))
14	13	anbi1d 737	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
15	14	rexbidv 3034	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
16	15	abbidv 2728	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})
17	16	mpteq2dv 4673	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
18	8, 17	mpteq12dv 4663	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
19		df-ldgis 36711	. . . . . . . 8 ⊢ ldgIdlSeq = (𝑟 ∈ V ↦ (𝑖 ∈ (LIdeal‘(Poly1‘𝑟)) ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((( deg1 ‘𝑟)‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
20		fvex 6113	. . . . . . . . . 10 ⊢ (LIdeal‘𝑃) ∈ V
21	7, 20	eqeltri 2684	. . . . . . . . 9 ⊢ 𝑈 ∈ V
22	21	mptex 6390	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})) ∈ V
23	18, 19, 22	fvmpt 6191	. . . . . . 7 ⊢ (𝑅 ∈ V → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
24	2, 23	syl 17	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → (ldgIdlSeq‘𝑅) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
25	1, 24	syl5eq 2656	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑆 = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})))
26	25	fveq1d 6105	. . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑆‘𝐼) = ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼))
27	26	fveq1d 6105	. . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑆‘𝐼)‘𝑋) = (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋))
28	27	3ad2ant1 1075	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋))
29		rexeq 3116	. . . . . . 7 ⊢ (𝑖 = 𝐼 → (∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))))
30	29	abbidv 2728	. . . . . 6 ⊢ (𝑖 = 𝐼 → {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})
31	30	mpteq2dv 4673	. . . . 5 ⊢ (𝑖 = 𝐼 → (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
32		eqid 2610	. . . . 5 ⊢ (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})) = (𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
33		nn0ex 11175	. . . . . 6 ⊢ ℕ0 ∈ V
34	33	mptex 6390	. . . . 5 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) ∈ V
35	31, 32, 34	fvmpt 6191	. . . 4 ⊢ (𝐼 ∈ 𝑈 → ((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))
36	35	fveq1d 6105	. . 3 ⊢ (𝐼 ∈ 𝑈 → (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})‘𝑋))
37	36	3ad2ant2 1076	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → (((𝑖 ∈ 𝑈 ↦ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝑖 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}))‘𝐼)‘𝑋) = ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})‘𝑋))
38		simp3 1056	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → 𝑋 ∈ ℕ0)
39		simpr 476	. . . . . 6 ⊢ (((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → 𝑗 = ((coe1‘𝑘)‘𝑋))
40	39	reximi 2994	. . . . 5 ⊢ (∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋)) → ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋))
41	40	ss2abi 3637	. . . 4 ⊢ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)}
42		abrexexg 7034	. . . . 5 ⊢ (𝐼 ∈ 𝑈 → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
43	42	3ad2ant2 1076	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V)
44		ssexg 4732	. . . 4 ⊢ (({𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ⊆ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∧ {𝑗 ∣ ∃𝑘 ∈ 𝐼 𝑗 = ((coe1‘𝑘)‘𝑋)} ∈ V) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ∈ V)
45	41, 43, 44	sylancr 694	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ∈ V)
46		breq2 4587	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝐷‘𝑘) ≤ 𝑥 ↔ (𝐷‘𝑘) ≤ 𝑋))
47		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((coe1‘𝑘)‘𝑥) = ((coe1‘𝑘)‘𝑋))
48	47	eqeq2d 2620	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑗 = ((coe1‘𝑘)‘𝑥) ↔ 𝑗 = ((coe1‘𝑘)‘𝑋)))
49	46, 48	anbi12d 743	. . . . . 6 ⊢ (𝑥 = 𝑋 → (((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))))
50	49	rexbidv 3034	. . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥)) ↔ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))))
51	50	abbidv 2728	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))} = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})
52		eqid 2610	. . . 4 ⊢ (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))}) = (𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})
53	51, 52	fvmptg 6189	. . 3 ⊢ ((𝑋 ∈ ℕ0 ∧ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))} ∈ V) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})
54	38, 45, 53	syl2anc 691	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑥 ∈ ℕ0 ↦ {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑥 ∧ 𝑗 = ((coe1‘𝑘)‘𝑥))})‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})
55	28, 37, 54	3eqtrd 2648	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑈 ∧ 𝑋 ∈ ℕ0) → ((𝑆‘𝐼)‘𝑋) = {𝑗 ∣ ∃𝑘 ∈ 𝐼 ((𝐷‘𝑘) ≤ 𝑋 ∧ 𝑗 = ((coe1‘𝑘)‘𝑋))})

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804   ≤ cle 9954  ℕ₀cn0 11169  LIdealclidl 18991  Poly₁cpl1 19368  coe₁cco1 19369   deg₁ cdg1 23618  ldgIdlSeqcldgis 36710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-nn 10898  df-n0 11170  df-ldgis 36711

This theorem is referenced by:  hbtlem2  36713  hbtlem4  36715  hbtlem3  36716  hbtlem5  36717  hbtlem6  36718