Theorem caucvgprlem2 7488

Description: Lemma for caucvgpr 7490. 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 7473	. . . 4 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
4		caucvgpr.f	. . . . 5 ⊢ (𝜑 → 𝐹:N⟶Q)
5		ltrelpi 7132	. . . . . . . 8 ⊢ <N ⊆ (N × N)
6	5	brel 4591	. . . . . . 7 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
7	1, 6	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐽 ∈ N ∧ 𝐾 ∈ N))
8	7	simprd 113	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ N)
9	4, 8	ffvelrnd 5556	. . . 4 ⊢ (𝜑 → (𝐹‘𝐾) ∈ Q)
10		ltanqi 7210	. . . 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 408	. . 3 ⊢ (𝜑 → ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q ((𝐹‘𝐾) +Q 𝑄))
12		ltbtwnnqq 7223	. . 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 121	. 2 ⊢ (𝜑 → ∃𝑥 ∈ Q (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))
14		simprl 520	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝑥 ∈ Q)
15	8	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝐾 ∈ N)
16		simprrl 528	. . . . . 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 5421	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
18		opeq1 3705	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
19	18	eceq1d 6465	. . . . . . . . . 10 ⊢ (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
20	19	fveq2d 5425	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
21	17, 20	oveq12d 5792	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
22	21	breq1d 3939	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥 ↔ ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥))
23	22	rspcev 2789	. . . . . 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 408	. . . . 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 3933	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
26	25	rexbidv 2438	. . . . . 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 5424	. . . . . . 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 7171	. . . . . . . . 9 ⊢ Q ∈ V
30	29	rabex 4072	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
31	29	rabex 4072	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
32	30, 31	op2nd 6045	. . . . . . 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 2160	. . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
34	26, 33	elrab2 2843	. . . . 5 ⊢ (𝑥 ∈ (2nd ‘𝐿) ↔ (𝑥 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
35	14, 24, 34	sylanbrc 413	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ (((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥 ∧ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))) → 𝑥 ∈ (2nd ‘𝐿))
36		simprrr 529	. . . . . 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 2689	. . . . . . 7 ⊢ 𝑥 ∈ V
38		breq1 3932	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q ((𝐹‘𝐾) +Q 𝑄) ↔ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))
39	37, 38	elab 2828	. . . . . 6 ⊢ (𝑥 ∈ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)} ↔ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄))
40	36, 39	sylibr 133	. . . . 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 7357	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)} ∈ V
42		gtnqex 7358	. . . . . 6 ⊢ {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢} ∈ V
43	41, 42	op1st 6044	. . . . 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 2233	. . . 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 2481	. . . 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 1214	. . 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 7484	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ P)
50	49	adantr 274	. . . 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 7183	. . . . . . 7 ⊢ (((𝐹‘𝐾) ∈ Q ∧ 𝑄 ∈ Q) → ((𝐹‘𝐾) +Q 𝑄) ∈ Q)
53	9, 51, 52	syl2anc 408	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐾) +Q 𝑄) ∈ Q)
54		nqprlu 7355	. . . . . 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 274	. . . 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 7314	. . . 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 408	. . 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 166	. 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 2554	1 ⊢ (𝜑 → 𝐿<P ⟨{𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)}, {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢}⟩)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  {crab 2420  ⟨cop 3530   class class class wbr 3929  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  1_oc1o 6306  [cec 6427  Ncnpi 7080   <_N clti 7083   ~_Q ceq 7087  Qcnq 7088   +_Q cplq 7090  *_Qcrq 7092   <_Q cltq 7093  Pcnp 7099  <_P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-inp 7274  df-iltp 7278

This theorem is referenced by:  caucvgprlemlim  7489