Theorem suplocexprlemdisj 7945

Description: Lemma for suplocexpr 7950. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)

Hypotheses

Ref	Expression
suplocexpr.m	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
suplocexpr.ub	⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
suplocexpr.loc	⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
suplocexpr.b	⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩

Assertion

Ref	Expression
suplocexprlemdisj	⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))

Distinct variable groups:   𝑤,𝐴,𝑢   𝑥,𝐴,𝑦   𝑤,𝐵   𝜑,𝑞,𝑤   𝜑,𝑥,𝑦   𝑢,𝑞

Allowed substitution hints:   𝜑(𝑧,𝑢)   𝐴(𝑧,𝑞)   𝐵(𝑥,𝑦,𝑧,𝑢,𝑞)

Proof of Theorem suplocexprlemdisj

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 531	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ ∪ (1st “ 𝐴))
2		suplocexprlemell 7938	. . . . 5 ⊢ (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
3	1, 2	sylib 122	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
4		simprr 533	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (1st ‘𝑠))
5		simplrr 538	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (2nd ‘𝐵))
6		suplocexpr.m	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
7		suplocexpr.ub	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥)
8		suplocexpr.loc	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑥 ∈ P ∀𝑦 ∈ P (𝑥<P 𝑦 → (∃𝑧 ∈ 𝐴 𝑥<P 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧<P 𝑦)))
9	6, 7, 8	suplocexprlemss 7940	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
10	9	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝐴 ⊆ P)
11		suplocexpr.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = ⟨∪ (1st “ 𝐴), {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}⟩
12	11	suplocexprlem2b 7939	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
13	12	eleq2d 2300	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ P → (𝑞 ∈ (2nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
14	10, 13	syl 14	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → (𝑞 ∈ (2nd ‘𝐵) ↔ 𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢}))
15		breq2 4093	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑞))
16	15	rexbidv 2532	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑞 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
17	16	elrab 2961	. . . . . . . . . 10 ⊢ (𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
18	14, 17	bitrdi 196	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → (𝑞 ∈ (2nd ‘𝐵) ↔ (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞)))
19	5, 18	mpbid 147	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
20	19	simprd 114	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞)
21		simprr 533	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑤 <Q 𝑞)
22	10	adantr 276	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → 𝐴 ⊆ P)
23		simplrl 537	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑠 ∈ 𝐴)
24	22, 23	sseldd 3227	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑠 ∈ P)
25		prop 7700	. . . . . . . . . 10 ⊢ (𝑠 ∈ P → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
26	24, 25	syl 14	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
27		eleq2 2294	. . . . . . . . . 10 ⊢ (𝑡 = (2nd ‘𝑠) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2nd ‘𝑠)))
28		simprl 531	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑤 ∈ ∩ (2nd “ 𝐴))
29		vex 2804	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
30	29	elint2 3936	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ∩ (2nd “ 𝐴) ↔ ∀𝑡 ∈ (2nd “ 𝐴)𝑤 ∈ 𝑡)
31	28, 30	sylib 122	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → ∀𝑡 ∈ (2nd “ 𝐴)𝑤 ∈ 𝑡)
32		fo2nd 6326	. . . . . . . . . . . . . 14 ⊢ 2nd :V–onto→V
33		fofun 5563	. . . . . . . . . . . . . 14 ⊢ (2nd :V–onto→V → Fun 2nd )
34	32, 33	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 2nd
35		vex 2804	. . . . . . . . . . . . . 14 ⊢ 𝑠 ∈ V
36		fof 5562	. . . . . . . . . . . . . . . 16 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
37	32, 36	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 2nd :V⟶V
38	37	fdmi 5492	. . . . . . . . . . . . . 14 ⊢ dom 2nd = V
39	35, 38	eleqtrri 2306	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ dom 2nd
40		funfvima 5891	. . . . . . . . . . . . 13 ⊢ ((Fun 2nd ∧ 𝑠 ∈ dom 2nd ) → (𝑠 ∈ 𝐴 → (2nd ‘𝑠) ∈ (2nd “ 𝐴)))
41	34, 39, 40	mp2an 426	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (2nd ‘𝑠) ∈ (2nd “ 𝐴))
42	41	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → (2nd ‘𝑠) ∈ (2nd “ 𝐴))
43	42	adantr 276	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → (2nd ‘𝑠) ∈ (2nd “ 𝐴))
44	27, 31, 43	rspcdva 2914	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑤 ∈ (2nd ‘𝑠))
45		prcunqu 7710	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝑠)) → (𝑤 <Q 𝑞 → 𝑞 ∈ (2nd ‘𝑠)))
46	26, 44, 45	syl2anc 411	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → (𝑤 <Q 𝑞 → 𝑞 ∈ (2nd ‘𝑠)))
47	21, 46	mpd 13	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) ∧ (𝑤 ∈ ∩ (2nd “ 𝐴) ∧ 𝑤 <Q 𝑞)) → 𝑞 ∈ (2nd ‘𝑠))
48	20, 47	rexlimddv 2654	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (2nd ‘𝑠))
49	4, 48	jca 306	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
50		simprl 531	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
51	10, 50	sseldd 3227	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ P)
52	51, 25	syl 14	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P)
53		simpllr 536	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ Q)
54		prdisj 7717	. . . . . 6 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
55	52, 53, 54	syl2anc 411	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
56	49, 55	pm2.21fal 1417	. . . 4 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ⊥)
57	3, 56	rexlimddv 2654	. . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → ⊥)
58	57	inegd 1416	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
59	58	ralrimiva 2604	1 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715   = wceq 1397  ⊥wfal 1402  ∃wex 1540   ∈ wcel 2201  ∀wral 2509  ∃wrex 2510  {crab 2513  Vcvv 2801   ⊆ wss 3199  ⟨cop 3673  ∪ cuni 3894  ∩ cint 3929   class class class wbr 4089  dom cdm 4727   “ cima 4730  Fun wfun 5322  ⟶wf 5324  –onto→wfo 5326  ‘cfv 5328  1^st c1st 6306  2^nd c2nd 6307  Qcnq 7505   <_Q cltq 7510  Pcnp 7516  <_P cltp 7520

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-1st 6308  df-2nd 6309  df-qs 6713  df-ni 7529  df-nqqs 7573  df-ltnqqs 7578  df-inp 7691  df-iltp 7695

This theorem is referenced by:  suplocexprlemex  7947