Theorem caucvgprprlemell 7486

Description: Lemma for caucvgprpr 7513. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)

Hypothesis

Ref	Expression
caucvgprprlemell.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩

Assertion

Ref	Expression
caucvgprprlemell	⊢ (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))

Distinct variable groups:   𝐹,𝑏   𝐹,𝑙,𝑟   𝑢,𝐹,𝑟   𝑋,𝑏,𝑝   𝑋,𝑙,𝑟,𝑝   𝑢,𝑋,𝑝   𝑋,𝑞,𝑏   𝑞,𝑙,𝑟   𝑢,𝑞

Allowed substitution hints:   𝐹(𝑞,𝑝)   𝐿(𝑢,𝑟,𝑞,𝑝,𝑏,𝑙)

Proof of Theorem caucvgprprlemell

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5774	. . . . . . . 8 ⊢ (𝑙 = 𝑋 → (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )))
2	1	breq2d 3936	. . . . . . 7 ⊢ (𝑙 = 𝑋 → (𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))))
3	2	abbidv 2255	. . . . . 6 ⊢ (𝑙 = 𝑋 → {𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))})
4	1	breq1d 3934	. . . . . . 7 ⊢ (𝑙 = 𝑋 → ((𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞))
5	4	abbidv 2255	. . . . . 6 ⊢ (𝑙 = 𝑋 → {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞})
6	3, 5	opeq12d 3708	. . . . 5 ⊢ (𝑙 = 𝑋 → ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩)
7	6	breq1d 3934	. . . 4 ⊢ (𝑙 = 𝑋 → (⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
8	7	rexbidv 2436	. . 3 ⊢ (𝑙 = 𝑋 → (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
9		caucvgprprlemell.lim	. . . . 5 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩
10	9	fveq2i 5417	. . . 4 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩)
11		nqex 7164	. . . . . 6 ⊢ Q ∈ V
12	11	rabex 4067	. . . . 5 ⊢ {𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)} ∈ V
13	11	rabex 4067	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩} ∈ V
14	12, 13	op1st 6037	. . . 4 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}, {𝑢 ∈ Q ∣ ∃𝑟 ∈ N ((𝐹‘𝑟) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝑟, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝑟, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑢}, {𝑞 ∣ 𝑢 <Q 𝑞}⟩}⟩) = {𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}
15	10, 14	eqtri 2158	. . 3 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑙 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)}
16	8, 15	elrab2 2838	. 2 ⊢ (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)))
17		opeq1 3700	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑎 → ⟨𝑟, 1o⟩ = ⟨𝑎, 1o⟩)
18	17	eceq1d 6458	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑎 → [⟨𝑟, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
19	18	fveq2d 5418	. . . . . . . . . 10 ⊢ (𝑟 = 𝑎 → (*Q‘[⟨𝑟, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
20	19	oveq2d 5783	. . . . . . . . 9 ⊢ (𝑟 = 𝑎 → (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
21	20	breq2d 3936	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))))
22	21	abbidv 2255	. . . . . . 7 ⊢ (𝑟 = 𝑎 → {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))})
23	20	breq1d 3934	. . . . . . . 8 ⊢ (𝑟 = 𝑎 → ((𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞))
24	23	abbidv 2255	. . . . . . 7 ⊢ (𝑟 = 𝑎 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞})
25	22, 24	opeq12d 3708	. . . . . 6 ⊢ (𝑟 = 𝑎 → ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩)
26		fveq2 5414	. . . . . 6 ⊢ (𝑟 = 𝑎 → (𝐹‘𝑟) = (𝐹‘𝑎))
27	25, 26	breq12d 3937	. . . . 5 ⊢ (𝑟 = 𝑎 → (⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎)))
28	27	cbvrexv 2653	. . . 4 ⊢ (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑎 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎))
29		opeq1 3700	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → ⟨𝑎, 1o⟩ = ⟨𝑏, 1o⟩)
30	29	eceq1d 6458	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → [⟨𝑎, 1o⟩] ~Q = [⟨𝑏, 1o⟩] ~Q )
31	30	fveq2d 5418	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝑏, 1o⟩] ~Q ))
32	31	oveq2d 5783	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) = (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )))
33	32	breq2d 3936	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))))
34	33	abbidv 2255	. . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))})
35	32	breq1d 3934	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞))
36	35	abbidv 2255	. . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞})
37	34, 36	opeq12d 3708	. . . . . 6 ⊢ (𝑎 = 𝑏 → ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩)
38		fveq2 5414	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
39	37, 38	breq12d 3937	. . . . 5 ⊢ (𝑎 = 𝑏 → (⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
40	39	cbvrexv 2653	. . . 4 ⊢ (∃𝑎 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑎, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑎) ↔ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
41	28, 40	bitri 183	. . 3 ⊢ (∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟) ↔ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏))
42	41	anbi2i 452	. 2 ⊢ ((𝑋 ∈ Q ∧ ∃𝑟 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑟, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑟)) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))
43	16, 42	bitri 183	1 ⊢ (𝑋 ∈ (1st ‘𝐿) ↔ (𝑋 ∈ Q ∧ ∃𝑏 ∈ N ⟨{𝑝 ∣ 𝑝 <Q (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑋 +Q (*Q‘[⟨𝑏, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝑏)))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2123  ∃wrex 2415  {crab 2418  ⟨cop 3525   class class class wbr 3924  ‘cfv 5118  (class class class)co 5767  1^st c1st 6029  1_oc1o 6299  [cec 6420  Ncnpi 7073   ~_Q ceq 7080  Qcnq 7081   +_Q cplq 7083  *_Qcrq 7085   <_Q cltq 7086   +_P cpp 7094  <_P cltp 7096

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 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-1st 6031  df-ec 6424  df-qs 6428  df-ni 7105  df-nqqs 7149

This theorem is referenced by:  caucvgprprlemopl  7498  caucvgprprlemlol  7499  caucvgprprlemdisj  7503  caucvgprprlemloc  7504