Theorem depindlem1 16346

Description: Lemma for depind 16349. (Contributed by Matthew House, 14-Apr-2026.)

Hypotheses

Ref	Expression
depind.p	⊢ (𝜑 → 𝑃:ℕ0⟶V)
depind.0	⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))
depind.h	⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))
depindlem1.4	⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))

Assertion

Ref	Expression
depindlem1	⊢ (𝜑 → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))

Distinct variable groups:   ℎ,𝑛,𝑥   𝐴,𝑚,𝑛   𝑛,𝐹   𝑚,𝐻,𝑛   𝑃,𝑛

Allowed substitution hints:   𝜑(𝑥,ℎ,𝑚,𝑛)   𝐴(𝑥,ℎ)   𝑃(𝑥,ℎ,𝑚)   𝐹(𝑥,ℎ,𝑚)   𝐻(𝑥,ℎ)

Proof of Theorem depindlem1

Dummy variables 𝑦 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0uz 9791	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
2		0zd 9491	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
3		nn0ex 9408	. . . . . . 7 ⊢ ℕ0 ∈ V
4	3	mptex 5880	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))) ∈ V
5		vex 2805	. . . . . 6 ⊢ 𝑦 ∈ V
6	4, 5	fvex 5659	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘𝑦) ∈ V
7	6	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘𝑦) ∈ V)
8		eqid 2231	. . . . . 6 ⊢ (𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)) = (𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))
9		vex 2805	. . . . . . 7 ⊢ ℎ ∈ V
10		vex 2805	. . . . . . 7 ⊢ 𝑥 ∈ V
11	9, 10	fvex 5659	. . . . . 6 ⊢ (ℎ‘𝑥) ∈ V
12		vex 2805	. . . . . 6 ⊢ 𝑧 ∈ V
13	8, 11, 5, 12	mpofvexi 6371	. . . . 5 ⊢ (𝑦(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))𝑧) ∈ V
14	13	a1i 9	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V)) → (𝑦(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))𝑧) ∈ V)
15	1, 2, 7, 14	seqf 10727	. . 3 ⊢ (𝜑 → seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))):ℕ0⟶V)
16		depindlem1.4	. . . 4 ⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))
17	16	feq1i 5475	. . 3 ⊢ (𝐹:ℕ0⟶V ↔ seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))):ℕ0⟶V)
18	15, 17	sylibr 134	. 2 ⊢ (𝜑 → 𝐹:ℕ0⟶V)
19	16	fveq1i 5640	. . . 4 ⊢ (𝐹‘0) = (seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))‘0)
20	6	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (ℤ≥‘0)) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘𝑦) ∈ V)
21	2, 20, 14	seq3-1 10725	. . . 4 ⊢ (𝜑 → (seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘0))
22	19, 21	eqtrid 2276	. . 3 ⊢ (𝜑 → (𝐹‘0) = ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘0))
23		0nn0 9417	. . . 4 ⊢ 0 ∈ ℕ0
24		depind.0	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))
25		iftrue 3610	. . . . 5 ⊢ (𝑚 = 0 → if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))) = 𝐴)
26		eqid 2231	. . . . 5 ⊢ (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))) = (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))
27	25, 26	fvmptg 5722	. . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ (𝑃‘0)) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘0) = 𝐴)
28	23, 24, 27	sylancr 414	. . 3 ⊢ (𝜑 → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘0) = 𝐴)
29	22, 28	eqtrd 2264	. 2 ⊢ (𝜑 → (𝐹‘0) = 𝐴)
30		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
31	30, 1	eleqtrdi 2324	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ≥‘0))
32	6	a1i 9	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑦 ∈ (ℤ≥‘0)) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘𝑦) ∈ V)
33	13	a1i 9	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ (𝑦 ∈ V ∧ 𝑧 ∈ V)) → (𝑦(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))𝑧) ∈ V)
34	31, 32, 33	seq3p1 10728	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))‘(𝑘 + 1)) = ((seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘(𝑘 + 1))))
35	16	fveq1i 5640	. . . . . . 7 ⊢ (𝐹‘(𝑘 + 1)) = (seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))‘(𝑘 + 1))
36	16	fveq1i 5640	. . . . . . . 8 ⊢ (𝐹‘𝑘) = (seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))‘𝑘)
37	36	oveq1i 6028	. . . . . . 7 ⊢ ((𝐹‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘(𝑘 + 1))) = ((seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘(𝑘 + 1)))
38	34, 35, 37	3eqtr4g 2289	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘(𝑘 + 1))))
39		eqeq1 2238	. . . . . . . . . 10 ⊢ (𝑚 = (𝑘 + 1) → (𝑚 = 0 ↔ (𝑘 + 1) = 0))
40		fvoveq1 6041	. . . . . . . . . 10 ⊢ (𝑚 = (𝑘 + 1) → (𝐻‘(𝑚 − 1)) = (𝐻‘((𝑘 + 1) − 1)))
41	39, 40	ifbieq2d 3630	. . . . . . . . 9 ⊢ (𝑚 = (𝑘 + 1) → if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))) = if((𝑘 + 1) = 0, 𝐴, (𝐻‘((𝑘 + 1) − 1))))
42		nn0p1nn 9441	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ0 → (𝑘 + 1) ∈ ℕ)
43	42	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ)
44	43	nnnn0d 9455	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
45	43	nnne0d 9188	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ≠ 0)
46	45	neneqd 2423	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ¬ (𝑘 + 1) = 0)
47	46	iffalsed 3615	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if((𝑘 + 1) = 0, 𝐴, (𝐻‘((𝑘 + 1) − 1))) = (𝐻‘((𝑘 + 1) − 1)))
48	30	nn0cnd 9457	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℂ)
49		pncan1 8556	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℂ → ((𝑘 + 1) − 1) = 𝑘)
50	48, 49	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑘 + 1) − 1) = 𝑘)
51	50	fveq2d 5643	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘((𝑘 + 1) − 1)) = (𝐻‘𝑘))
52	47, 51	eqtrd 2264	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if((𝑘 + 1) = 0, 𝐴, (𝐻‘((𝑘 + 1) − 1))) = (𝐻‘𝑘))
53		depind.h	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))
54		fveq2 5639	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐻‘𝑛) = (𝐻‘𝑘))
55		fveq2 5639	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑃‘𝑛) = (𝑃‘𝑘))
56		fvoveq1 6041	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑃‘(𝑛 + 1)) = (𝑃‘(𝑘 + 1)))
57	54, 55, 56	feq123d 5473	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)) ↔ (𝐻‘𝑘):(𝑃‘𝑘)⟶(𝑃‘(𝑘 + 1))))
58	57	rspccva 2909	. . . . . . . . . . . 12 ⊢ ((∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)) ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘):(𝑃‘𝑘)⟶(𝑃‘(𝑘 + 1)))
59	53, 58	sylan 283	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘):(𝑃‘𝑘)⟶(𝑃‘(𝑘 + 1)))
60		depind.p	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃:ℕ0⟶V)
61	60	ffvelcdmda 5782	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝑃‘𝑘) ∈ V)
62	59, 61	fexd 5884	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) ∈ V)
63	52, 62	eqeltrd 2308	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → if((𝑘 + 1) = 0, 𝐴, (𝐻‘((𝑘 + 1) − 1))) ∈ V)
64	26, 41, 44, 63	fvmptd3 5740	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘(𝑘 + 1)) = if((𝑘 + 1) = 0, 𝐴, (𝐻‘((𝑘 + 1) − 1))))
65	64, 52	eqtrd 2264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘(𝑘 + 1)) = (𝐻‘𝑘))
66	65	oveq2d 6034	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐹‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))((𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1))))‘(𝑘 + 1))) = ((𝐹‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))(𝐻‘𝑘)))
67	38, 66	eqtrd 2264	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐹‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))(𝐻‘𝑘)))
68	18	ffvelcdmda 5782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ V)
69		fvexg 5658	. . . . . . 7 ⊢ (((𝐻‘𝑘) ∈ V ∧ (𝐹‘𝑘) ∈ V) → ((𝐻‘𝑘)‘(𝐹‘𝑘)) ∈ V)
70	62, 68, 69	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐻‘𝑘)‘(𝐹‘𝑘)) ∈ V)
71		fveq2 5639	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑘) → (𝑧‘𝑦) = (𝑧‘(𝐹‘𝑘)))
72		fveq1 5638	. . . . . . 7 ⊢ (𝑧 = (𝐻‘𝑘) → (𝑧‘(𝐹‘𝑘)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
73		fveq2 5639	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (ℎ‘𝑥) = (ℎ‘𝑦))
74		fveq1 5638	. . . . . . . 8 ⊢ (ℎ = 𝑧 → (ℎ‘𝑦) = (𝑧‘𝑦))
75	73, 74	cbvmpov 6101	. . . . . . 7 ⊢ (𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)) = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑧‘𝑦))
76	71, 72, 75	ovmpog 6156	. . . . . 6 ⊢ (((𝐹‘𝑘) ∈ V ∧ (𝐻‘𝑘) ∈ V ∧ ((𝐻‘𝑘)‘(𝐹‘𝑘)) ∈ V) → ((𝐹‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))(𝐻‘𝑘)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
77	68, 62, 70, 76	syl3anc 1273	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐹‘𝑘)(𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥))(𝐻‘𝑘)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
78	67, 77	eqtrd 2264	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
79	78	ralrimiva 2605	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
80		fvoveq1 6041	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
81		fveq2 5639	. . . . . 6 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
82	54, 81	fveq12d 5646	. . . . 5 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛)‘(𝐹‘𝑛)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
83	80, 82	eqeq12d 2246	. . . 4 ⊢ (𝑛 = 𝑘 → ((𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ↔ (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘))))
84	83	cbvralvw 2771	. . 3 ⊢ (∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ↔ ∀𝑘 ∈ ℕ0 (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
85	79, 84	sylibr 134	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)))
86	18, 29, 85	3jca 1203	1 ⊢ (𝜑 → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ifcif 3605   ↦ cmpt 4150  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018   ∈ cmpo 6020  ℂcc 8030  0cc0 8032  1c1 8033   + caddc 8035   − cmin 8350  ℕcn 9143  ℕ₀cn0 9402  ℤ_≥cuz 9755  seqcseq 10710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-seqfrec 10711

This theorem is referenced by:  depindlem2  16347  depindlem3  16348  depind  16349