Theorem caucvgprlem2 7943

Description: Lemma for caucvgpr 7945. 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 7928	. . . 4 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑄)
4		caucvgpr.f	. . . . 5 ⊢ (𝜑 → 𝐹:N⟶Q)
5		ltrelpi 7587	. . . . . . . 8 ⊢ <N ⊆ (N × N)
6	5	brel 4784	. . . . . . 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 5791	. . . 4 ⊢ (𝜑 → (𝐹‘𝐾) ∈ Q)
10		ltanqi 7665	. . . 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 7678	. . 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 531	. . . 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 541	. . . . . 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 5648	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
18		opeq1 3867	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐾 → ⟨𝑗, 1o⟩ = ⟨𝐾, 1o⟩)
19	18	eceq1d 6781	. . . . . . . . . 10 ⊢ (𝑗 = 𝐾 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐾, 1o⟩] ~Q )
20	19	fveq2d 5652	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝐾, 1o⟩] ~Q ))
21	17, 20	oveq12d 6046	. . . . . . . 8 ⊢ (𝑗 = 𝐾 → ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )))
22	21	breq1d 4103	. . . . . . 7 ⊢ (𝑗 = 𝐾 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥 ↔ ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑥))
23	22	rspcev 2911	. . . . . 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 4097	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑥))
26	25	rexbidv 2534	. . . . . 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 5651	. . . . . . 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 7626	. . . . . . . . 9 ⊢ Q ∈ V
30	29	rabex 4239	. . . . . . . 8 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
31	29	rabex 4239	. . . . . . . 8 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
32	30, 31	op2nd 6319	. . . . . . 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 2252	. . . . . 6 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
34	26, 33	elrab2 2966	. . . . 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 542	. . . . . 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 2806	. . . . . . 7 ⊢ 𝑥 ∈ V
38		breq1 4096	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q ((𝐹‘𝐾) +Q 𝑄) ↔ 𝑥 <Q ((𝐹‘𝐾) +Q 𝑄)))
39	37, 38	elab 2951	. . . . . 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 7812	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q ((𝐹‘𝐾) +Q 𝑄)} ∈ V
42		gtnqex 7813	. . . . . 6 ⊢ {𝑢 ∣ ((𝐹‘𝐾) +Q 𝑄) <Q 𝑢} ∈ V
43	41, 42	op1st 6318	. . . . 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 2325	. . . 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 2582	. . . 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 1272	. . 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 7939	. . . . 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 7638	. . . . . . 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 7810	. . . . . 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 7769	. . . 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 2656	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 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  {crab 2515  ⟨cop 3676   class class class wbr 4093  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028  1^st c1st 6310  2^nd c2nd 6311  1_oc1o 6618  [cec 6743  Ncnpi 7535   <_N clti 7538   ~_Q ceq 7542  Qcnq 7543   +_Q cplq 7545  *_Qcrq 7547   <_Q cltq 7548  Pcnp 7554  <_P cltp 7558

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-inp 7729  df-iltp 7733

This theorem is referenced by:  caucvgprlemlim  7944