Theorem caucvgprlem2 7657

Description: Lemma for caucvgpr 7659. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
caucvgprlemlim.q	⊢ (𝜑 → 𝑄 ∈ Q)
caucvgprlemlim.jk	⊢ (𝜑 → 𝐽 <N 𝐾)
caucvgprlemlim.jkq	⊢ (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)

Assertion

Ref	Expression
caucvgprlem2	⊢ (𝜑 → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩)

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑢,𝑙   𝑛,𝐹,𝑘   𝑗,𝐾,𝑢,𝑙   𝑗,𝐿,𝑘   𝑄,𝑙,𝑢   𝑗,𝑙   𝑗,𝑘   𝑘,𝑛

Allowed substitution hints:   𝜑(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝑄(𝑗,𝑘,𝑛)   𝐽(𝑢,𝑗,𝑘,𝑛,𝑙)   𝐾(𝑘,𝑛)   𝐿(𝑢,𝑛,𝑙)

Proof of Theorem caucvgprlem2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprlemlim.jk	. . . . 5 ⊢ (𝜑 → 𝐽 <N 𝐾)
2		caucvgprlemlim.jkq	. . . . 5 ⊢ (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑄)
3	1, 2	caucvgprlemk 7642	. . . 4 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
4		caucvgpr.f	. . . . 5 ⊢ (𝜑 → 𝐹:N⟶Q)
5		ltrelpi 7301	. . . . . . . 8 ⊢ <N ⊆ (N × N)
6	5	brel 4674	. . . . . . 7 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
7	1, 6	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
8	7	simprd 114	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ N)
9	4, 8	ffvelcdmd 5647	. . . 4 ⊢ (𝜑 → (𝐹‘𝐾) ∈ Q)
10		ltanqi 7379	. . . 4 ⊢ (((*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄 ∧ (𝐹‘𝐾) ∈ Q) → ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄))
11	3, 9, 10	syl2anc 411	. . 3 ⊢ (𝜑 → ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄))
12		ltbtwnnqq 7392	. . 3 ⊢ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄) ↔ ∃𝑥 ∈ Q (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))
13	11, 12	sylib 122	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Q (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))
14		simprl 529	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝑥 ∈ Q)
15	8	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝐾 ∈ N)
16		simprrl 539	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥)
17		fveq2 5510	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
18		opeq1 3776	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
19	18	eceq1d 6564	. . . . . . . . . 10 ⊢ (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
20	19	fveq2d 5514	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
21	17, 20	oveq12d 5886	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
22	21	breq1d 4010	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥 ↔ ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥))
23	22	rspcev 2841	. . . . . 6 ⊢ ((𝐾 ∈ N ∧ ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥)
24	15, 16, 23	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥)
25		breq2 4004	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
26	25	rexbidv 2478	. . . . . 6 ⊢ (𝑢 = 𝑥 → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
27		caucvgpr.lim	. . . . . . . 8 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
28	27	fveq2i 5513	. . . . . . 7 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
29		nqex 7340	. . . . . . . . 9 ⊢ Q ∈ V
30	29	rabex 4144	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
31	29	rabex 4144	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
32	30, 31	op2nd 6141	. . . . . . 7 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
33	28, 32	eqtri 2198	. . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
34	26, 33	elrab2 2896	. . . . 5 ⊢ (𝑥 ∈ (2nd ‘𝐿) ↔ (𝑥 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
35	14, 24, 34	sylanbrc 417	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝑥 ∈ (2nd ‘𝐿))
36		simprrr 540	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄))
37		vex 2740	. . . . . . 7 ⊢ 𝑥 ∈ V
38		breq1 4003	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q ((𝐹‘𝐾) +Q 𝑄) ↔ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))
39	37, 38	elab 2881	. . . . . 6 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)} ↔ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄))
40	36, 39	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝑥 ∈ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)})
41		ltnqex 7526	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)} ∈ V
42		gtnqex 7527	. . . . . 6 ⊢ {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢} ∈ V
43	41, 42	op1st 6140	. . . . 5 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}
44	40, 43	eleqtrrdi 2271	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩))
45		rspe 2526	. . . 4 ⊢ ((𝑥 ∈ Q ∧ (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩)))
46	14, 35, 44, 45	syl12anc 1236	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩)))
47		caucvgpr.cau	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
48		caucvgpr.bnd	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
49	4, 47, 48, 27	caucvgprlemcl 7653	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ P)
50	49	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝐿 ∈ P)
51		caucvgprlemlim.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Q)
52		addclnq 7352	. . . . . . 7 ⊢ (((𝐹‘𝐾) ∈ Q ∧ 𝑄 ∈ Q) → ((𝐹‘𝐾) +Q 𝑄) ∈ Q)
53	9, 51, 52	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐾) +Q 𝑄) ∈ Q)
54		nqprlu 7524	. . . . . 6 ⊢ (((𝐹‘𝐾) +Q 𝑄) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P)
55	53, 54	syl 14	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P)
56	55	adantr 276	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P)
57		ltdfpr 7483	. . . 4 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩ ∈ P) → (𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩))))
58	50, 56, 57	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → (𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩))))
59	46, 58	mpbird 167	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩)
60	13, 59	rexlimddv 2599	1 ⊢ (𝜑 → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  {crab 2459  ⟨cop 3594   class class class wbr 4000  ⟶wf 5207  ‘cfv 5211  (class class class)co 5868  1^st c1st 6132  2^nd c2nd 6133  1_oc1o 6403  [cec 6526  Ncnpi 7249   <_N clti 7252   ~_Q ceq 7256  Qcnq 7257   +_Q cplq 7259  *_Qcrq 7261   <_Q cltq 7262  Pcnp 7268  <_P cltp 7272

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4285  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-lti 7284  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-1nqqs 7328  df-rq 7329  df-ltnqqs 7330  df-inp 7443  df-iltp 7447

This theorem is referenced by:  caucvgprlemlim  7658