Theorem suplocexprlemdisj 7782

Description: Lemma for suplocexpr 7787. 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 529	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ ∪ (1st “ 𝐴))
2		suplocexprlemell 7775	. . . . 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 531	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (1st ‘𝑠))
5		simplrr 536	. . . . . . . . 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 7777	. . . . . . . . . . . 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 7776	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
13	12	eleq2d 2263	. . . . . . . . . . 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 4034	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑞))
16	15	rexbidv 2495	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑞 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
17	16	elrab 2917	. . . . . . . . . 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 531	. . . . . . . 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 535	. . . . . . . . . . 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 3181	. . . . . . . . . 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 7537	. . . . . . . . . 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 2257	. . . . . . . . . 10 ⊢ (𝑡 = (2nd ‘𝑠) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2nd ‘𝑠)))
28		simprl 529	. . . . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
30	29	elint2 3878	. . . . . . . . . . 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 6213	. . . . . . . . . . . . . 14 ⊢ 2nd :V–onto→V
33		fofun 5478	. . . . . . . . . . . . . 14 ⊢ (2nd :V–onto→V → Fun 2nd )
34	32, 33	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 2nd
35		vex 2763	. . . . . . . . . . . . . 14 ⊢ 𝑠 ∈ V
36		fof 5477	. . . . . . . . . . . . . . . 16 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
37	32, 36	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 2nd :V⟶V
38	37	fdmi 5412	. . . . . . . . . . . . . 14 ⊢ dom 2nd = V
39	35, 38	eleqtrri 2269	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ dom 2nd
40		funfvima 5791	. . . . . . . . . . . . 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 2870	. . . . . . . . 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 7547	. . . . . . . . 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 2616	. . . . . 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 529	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
51	10, 50	sseldd 3181	. . . . . . 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 534	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ Q)
54		prdisj 7554	. . . . . 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 1384	. . . 4 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ⊥)
57	3, 56	rexlimddv 2616	. . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → ⊥)
58	57	inegd 1383	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
59	58	ralrimiva 2567	1 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   = wceq 1364  ⊥wfal 1369  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  {crab 2476  Vcvv 2760   ⊆ wss 3154  ⟨cop 3622  ∪ cuni 3836  ∩ cint 3871   class class class wbr 4030  dom cdm 4660   “ cima 4663  Fun wfun 5249  ⟶wf 5251  –onto→wfo 5253  ‘cfv 5255  1^st c1st 6193  2^nd c2nd 6194  Qcnq 7342   <_Q cltq 7347  Pcnp 7353  <_P cltp 7357

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6195  df-2nd 6196  df-qs 6595  df-ni 7366  df-nqqs 7410  df-ltnqqs 7415  df-inp 7528  df-iltp 7532

This theorem is referenced by:  suplocexprlemex  7784