Theorem depindlem3 16490

Description: Lemma for depind 16491. (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
depindlem3	⊢ (𝜑 → ∀𝑓 ∈ X 𝑛 ∈ ℕ0 (𝑃‘𝑛)(((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))

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

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

Proof of Theorem depindlem3

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

Step	Hyp	Ref	Expression
1		ixpfn 6938	. . . . 5 ⊢ (𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛) → 𝑓 Fn ℕ0)
2	1	ad2antlr 489	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝑓 Fn ℕ0)
3		depind.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃:ℕ0⟶V)
4		depind.0	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝑃‘0))
5		depind.h	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐻‘𝑛):(𝑃‘𝑛)⟶(𝑃‘(𝑛 + 1)))
6		depindlem1.4	. . . . . . . 8 ⊢ 𝐹 = seq0((𝑥 ∈ V, ℎ ∈ V ↦ (ℎ‘𝑥)), (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐴, (𝐻‘(𝑚 − 1)))))
7	3, 4, 5, 6	depindlem1 16488	. . . . . . 7 ⊢ (𝜑 → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))
8	7	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝐹:ℕ0⟶V ∧ (𝐹‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛))))
9	8	simp1d 1036	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝐹:ℕ0⟶V)
10	9	ffnd 5508	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝐹 Fn ℕ0)
11		fveq2 5669	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝑓‘𝑦) = (𝑓‘0))
12		fveq2 5669	. . . . . . . 8 ⊢ (𝑦 = 0 → (𝐹‘𝑦) = (𝐹‘0))
13	11, 12	eqeq12d 2247	. . . . . . 7 ⊢ (𝑦 = 0 → ((𝑓‘𝑦) = (𝐹‘𝑦) ↔ (𝑓‘0) = (𝐹‘0)))
14	13	imbi2d 230	. . . . . 6 ⊢ (𝑦 = 0 → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑦) = (𝐹‘𝑦)) ↔ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘0) = (𝐹‘0))))
15		fveq2 5669	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (𝑓‘𝑦) = (𝑓‘𝑘))
16		fveq2 5669	. . . . . . . 8 ⊢ (𝑦 = 𝑘 → (𝐹‘𝑦) = (𝐹‘𝑘))
17	15, 16	eqeq12d 2247	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ((𝑓‘𝑦) = (𝐹‘𝑦) ↔ (𝑓‘𝑘) = (𝐹‘𝑘)))
18	17	imbi2d 230	. . . . . 6 ⊢ (𝑦 = 𝑘 → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑦) = (𝐹‘𝑦)) ↔ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑘) = (𝐹‘𝑘))))
19		fveq2 5669	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (𝑓‘𝑦) = (𝑓‘(𝑘 + 1)))
20		fveq2 5669	. . . . . . . 8 ⊢ (𝑦 = (𝑘 + 1) → (𝐹‘𝑦) = (𝐹‘(𝑘 + 1)))
21	19, 20	eqeq12d 2247	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ((𝑓‘𝑦) = (𝐹‘𝑦) ↔ (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1))))
22	21	imbi2d 230	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑦) = (𝐹‘𝑦)) ↔ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
23		simprl 531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘0) = 𝐴)
24	8	simp2d 1037	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝐹‘0) = 𝐴)
25	23, 24	eqtr4d 2268	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘0) = (𝐹‘0))
26		fveq2 5669	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑘) = (𝐹‘𝑘) → ((𝐻‘𝑘)‘(𝑓‘𝑘)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
27	26	ad2antlr 489	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → ((𝐻‘𝑘)‘(𝑓‘𝑘)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
28		simplrr 538	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) → ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))
29		fvoveq1 6072	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝑓‘(𝑛 + 1)) = (𝑓‘(𝑘 + 1)))
30		fveq2 5669	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐻‘𝑛) = (𝐻‘𝑘))
31		fveq2 5669	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
32	30, 31	fveq12d 5676	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛)‘(𝑓‘𝑛)) = ((𝐻‘𝑘)‘(𝑓‘𝑘)))
33	29, 32	eqeq12d 2247	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)) ↔ (𝑓‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝑓‘𝑘))))
34	33	rspccva 2919	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝑓‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝑓‘𝑘)))
35	28, 34	sylan 283	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → (𝑓‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝑓‘𝑘)))
36	8	simp3d 1038	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)))
37	36	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) → ∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)))
38		fvoveq1 6072	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
39		fveq2 5669	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
40	30, 39	fveq12d 5676	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑘 → ((𝐻‘𝑛)‘(𝐹‘𝑛)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
41	38, 40	eqeq12d 2247	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → ((𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ↔ (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘))))
42	41	rspccva 2919	. . . . . . . . . . 11 ⊢ ((∀𝑛 ∈ ℕ0 (𝐹‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
43	37, 42	sylan 283	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → (𝐹‘(𝑘 + 1)) = ((𝐻‘𝑘)‘(𝐹‘𝑘)))
44	27, 35, 43	3eqtr4d 2275	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ (𝑓‘𝑘) = (𝐹‘𝑘)) ∧ 𝑘 ∈ ℕ0) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))
45	44	exp31 364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → ((𝑓‘𝑘) = (𝐹‘𝑘) → (𝑘 ∈ ℕ0 → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
46	45	com3r 79	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → ((𝑓‘𝑘) = (𝐹‘𝑘) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
47	46	a2d 26	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑘) = (𝐹‘𝑘)) → (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘(𝑘 + 1)) = (𝐹‘(𝑘 + 1)))))
48	14, 18, 22, 18, 25, 47	nn0ind 9688	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → (𝑓‘𝑘) = (𝐹‘𝑘)))
49	48	impcom 125	. . . 4 ⊢ ((((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) ∧ 𝑘 ∈ ℕ0) → (𝑓‘𝑘) = (𝐹‘𝑘))
50	2, 10, 49	eqfnfvd 5777	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) ∧ ((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛)))) → 𝑓 = 𝐹)
51	50	ex 115	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ X𝑛 ∈ ℕ0 (𝑃‘𝑛)) → (((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))
52	51	ralrimiva 2615	1 ⊢ (𝜑 → ∀𝑓 ∈ X 𝑛 ∈ ℕ0 (𝑃‘𝑛)(((𝑓‘0) = 𝐴 ∧ ∀𝑛 ∈ ℕ0 (𝑓‘(𝑛 + 1)) = ((𝐻‘𝑛)‘(𝑓‘𝑛))) → 𝑓 = 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2812  ifcif 3619   ↦ cmpt 4170   Fn wfn 5346  ⟶wf 5347  ‘cfv 5351  (class class class)co 6049   ∈ cmpo 6051  Xcixp 6932  0cc0 8123  1c1 8124   + caddc 8126   − cmin 8440  ℕ₀cn0 9492  seqcseq 10805

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-addass 8225  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-ltadd 8239

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-ixp 6933  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-seqfrec 10806

This theorem is referenced by:  depind  16491