Theorem suplocexprlemdisj 7661

Description: Lemma for suplocexpr 7666. 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 521	. . . . 5 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → 𝑞 ∈ ∪ (1st “ 𝐴))
2		suplocexprlemell 7654	. . . . 5 ⊢ (𝑞 ∈ ∪ (1st “ 𝐴) ↔ ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
3	1, 2	sylib 121	. . . 4 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → ∃𝑠 ∈ 𝐴 𝑞 ∈ (1st ‘𝑠))
4		simprr 522	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (1st ‘𝑠))
5		simplrr 526	. . . . . . . . 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 7656	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ P)
10	9	ad3antrrr 484	. . . . . . . . . . 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 7655	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ P → (2nd ‘𝐵) = {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢})
13	12	eleq2d 2236	. . . . . . . . . . 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 3986	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑞 → (𝑤 <Q 𝑢 ↔ 𝑤 <Q 𝑞))
16	15	rexbidv 2467	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑞 → (∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢 ↔ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
17	16	elrab 2882	. . . . . . . . . 10 ⊢ (𝑞 ∈ {𝑢 ∈ Q ∣ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑢} ↔ (𝑞 ∈ Q ∧ ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞))
18	14, 17	bitrdi 195	. . . . . . . . 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 146	. . . . . . . 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 113	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ∃𝑤 ∈ ∩ (2nd “ 𝐴)𝑤 <Q 𝑞)
21		simprr 522	. . . . . . . 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 274	. . . . . . . . . . 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 525	. . . . . . . . . . 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 3143	. . . . . . . . . 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 7416	. . . . . . . . . 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 2230	. . . . . . . . . 10 ⊢ (𝑡 = (2nd ‘𝑠) → (𝑤 ∈ 𝑡 ↔ 𝑤 ∈ (2nd ‘𝑠)))
28		simprl 521	. . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ 𝑤 ∈ V
30	29	elint2 3831	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ∩ (2nd “ 𝐴) ↔ ∀𝑡 ∈ (2nd “ 𝐴)𝑤 ∈ 𝑡)
31	28, 30	sylib 121	. . . . . . . . . 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 6126	. . . . . . . . . . . . . 14 ⊢ 2nd :V–onto→V
33		fofun 5411	. . . . . . . . . . . . . 14 ⊢ (2nd :V–onto→V → Fun 2nd )
34	32, 33	ax-mp 5	. . . . . . . . . . . . 13 ⊢ Fun 2nd
35		vex 2729	. . . . . . . . . . . . . 14 ⊢ 𝑠 ∈ V
36		fof 5410	. . . . . . . . . . . . . . . 16 ⊢ (2nd :V–onto→V → 2nd :V⟶V)
37	32, 36	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 2nd :V⟶V
38	37	fdmi 5345	. . . . . . . . . . . . . 14 ⊢ dom 2nd = V
39	35, 38	eleqtrri 2242	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ dom 2nd
40		funfvima 5716	. . . . . . . . . . . . 13 ⊢ ((Fun 2nd ∧ 𝑠 ∈ dom 2nd ) → (𝑠 ∈ 𝐴 → (2nd ‘𝑠) ∈ (2nd “ 𝐴)))
41	34, 39, 40	mp2an 423	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ 𝐴 → (2nd ‘𝑠) ∈ (2nd “ 𝐴))
42	41	ad2antrl 482	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → (2nd ‘𝑠) ∈ (2nd “ 𝐴))
43	42	adantr 274	. . . . . . . . . 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 2835	. . . . . . . . 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 7426	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑤 ∈ (2nd ‘𝑠)) → (𝑤 <Q 𝑞 → 𝑞 ∈ (2nd ‘𝑠)))
46	26, 44, 45	syl2anc 409	. . . . . . . 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 2588	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ (2nd ‘𝑠))
49	4, 48	jca 304	. . . . 5 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
50		simprl 521	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑠 ∈ 𝐴)
51	10, 50	sseldd 3143	. . . . . . 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 524	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → 𝑞 ∈ Q)
54		prdisj 7433	. . . . . 6 ⊢ ((⟨(1st ‘𝑠), (2nd ‘𝑠)⟩ ∈ P ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ (1st ‘𝑠) ∧ 𝑞 ∈ (2nd ‘𝑠)))
55	52, 53, 54	syl2anc 409	. . . . 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 1363	. . . 4 ⊢ ((((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) ∧ (𝑠 ∈ 𝐴 ∧ 𝑞 ∈ (1st ‘𝑠))) → ⊥)
57	3, 56	rexlimddv 2588	. . 3 ⊢ (((𝜑 ∧ 𝑞 ∈ Q) ∧ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵))) → ⊥)
58	57	inegd 1362	. 2 ⊢ ((𝜑 ∧ 𝑞 ∈ Q) → ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))
59	58	ralrimiva 2539	1 ⊢ (𝜑 → ∀𝑞 ∈ Q ¬ (𝑞 ∈ ∪ (1st “ 𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343  ⊥wfal 1348  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  {crab 2448  Vcvv 2726   ⊆ wss 3116  ⟨cop 3579  ∪ cuni 3789  ∩ cint 3824   class class class wbr 3982  dom cdm 4604   “ cima 4607  Fun wfun 5182  ⟶wf 5184  –onto→wfo 5186  ‘cfv 5188  1^st c1st 6106  2^nd c2nd 6107  Qcnq 7221   <_Q cltq 7226  Pcnp 7232  <_P cltp 7236

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-qs 6507  df-ni 7245  df-nqqs 7289  df-ltnqqs 7294  df-inp 7407  df-iltp 7411

This theorem is referenced by:  suplocexprlemex  7663