Theorem ordtrest2NEWlem 33891

Description: Lemma for ordtrest2NEW 33892. (Contributed by Mario Carneiro, 9-Sep-2015.) (Revised by Thierry Arnoux, 11-Sep-2018.)

Hypotheses

Ref	Expression
ordtNEW.b	⊢ 𝐵 = (Base‘𝐾)
ordtNEW.l	⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
ordtrest2NEW.2	⊢ (𝜑 → 𝐾 ∈ Toset)
ordtrest2NEW.3	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
ordtrest2NEW.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)

Assertion

Ref	Expression
ordtrest2NEWlem	⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))

Distinct variable groups:   𝑥,𝑦, ≤   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦   𝑥,𝐴,𝑦,𝑣,𝑤,𝑧   𝑣, ≤   𝑥,𝑤,𝑧,𝑦, ≤   𝑣,𝐴,𝑤,𝑧   𝑣,𝐵,𝑤,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑤,𝑣)   𝐾(𝑧,𝑤,𝑣)

Proof of Theorem ordtrest2NEWlem

Step	Hyp	Ref	Expression
1		inrab2 4270	. . . . 5 ⊢ ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) = {𝑤 ∈ (𝐵 ∩ 𝐴) ∣ ¬ 𝑤 ≤ 𝑧}
2		ordtrest2NEW.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		sseqin2 4176	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
4	2, 3	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐵 ∩ 𝐴) = 𝐴)
6		rabeq 3411	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) = 𝐴 → {𝑤 ∈ (𝐵 ∩ 𝐴) ∣ ¬ 𝑤 ≤ 𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧})
7	5, 6	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → {𝑤 ∈ (𝐵 ∩ 𝐴) ∣ ¬ 𝑤 ≤ 𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧})
8	1, 7	eqtrid 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧})
9		ordtNEW.l	. . . . . . . . . . . . 13 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
10		fvex 6839	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) ∈ V
11	10	inex1 5259	. . . . . . . . . . . . 13 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
12	9, 11	eqeltri 2824	. . . . . . . . . . . 12 ⊢ ≤ ∈ V
13	12	inex1 5259	. . . . . . . . . . 11 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) ∈ V)
15		eqid 2729	. . . . . . . . . . 11 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
16	15	ordttopon 23096	. . . . . . . . . 10 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))))
18		ordtrest2NEW.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Toset)
19		tospos 18342	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
20		posprs 18240	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Poset → 𝐾 ∈ Proset )
21	18, 19, 20	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ Proset )
22		ordtNEW.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐾)
23	22, 9	prsssdm 33886	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
24	21, 2, 23	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
25	24	fveq2d 6830	. . . . . . . . 9 ⊢ (𝜑 → (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
26	17, 25	eleqtrd 2830	. . . . . . . 8 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
27		toponmax 22829	. . . . . . . 8 ⊢ ((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
30		rabid2 3430	. . . . . . 7 ⊢ (𝐴 = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
31		eleq1 2816	. . . . . . 7 ⊢ (𝐴 = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} → (𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
32	30, 31	sylbir 235	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 → (𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
33	29, 32	syl5ibcom 245	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
34		dfrex2 3056	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 𝑤 ≤ 𝑧 ↔ ¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
35		breq1 5098	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ≤ 𝑧 ↔ 𝑥 ≤ 𝑧))
36	35	cbvrexvw 3208	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 𝑤 ≤ 𝑧 ↔ ∃𝑥 ∈ 𝐴 𝑥 ≤ 𝑧)
37	34, 36	bitr3i 277	. . . . . 6 ⊢ (¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 ↔ ∃𝑥 ∈ 𝐴 𝑥 ≤ 𝑧)
38		ordttop 23103	. . . . . . . . . . . . 13 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
39	14, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
41		0opn 22807	. . . . . . . . . . 11 ⊢ ((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top → ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
43	42	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
44		eleq1 2816	. . . . . . . . 9 ⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} = ∅ → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
45	43, 44	syl5ibrcom 247	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} = ∅ → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
46		rabn0 4342	. . . . . . . . . 10 ⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
47		breq1 5098	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧))
48	47	notbid 318	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (¬ 𝑤 ≤ 𝑧 ↔ ¬ 𝑦 ≤ 𝑧))
49	48	cbvrexvw 3208	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑧)
50	46, 49	bitri 275	. . . . . . . . 9 ⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ≠ ∅ ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑧)
51	18	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝐾 ∈ Toset)
52	2	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → 𝐴 ⊆ 𝐵)
53	52	sselda 3937	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
54		simpllr 775	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐵)
55	22, 9	trleile 32926	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ≤ 𝑧 ∨ 𝑧 ≤ 𝑦))
56	51, 53, 54, 55	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑧 ∨ 𝑧 ≤ 𝑦))
57	56	ord 864	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑧 → 𝑧 ≤ 𝑦))
58		an4 656	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)))
59		ordtrest2NEW.4	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)
60		rabss 4025	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
61	59, 60	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
62	61	r19.21bi 3221	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
63	62	an32s 652	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
64	63	impr 454	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))) → 𝑧 ∈ 𝐴)
65	58, 64	sylan2b 594	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → 𝑧 ∈ 𝐴)
66		brinxp 5702	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑤 ≤ 𝑧 ↔ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧))
67	66	ancoms 458	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝑧 ↔ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧))
68	67	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 ≤ 𝑧 ↔ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧))
69	68	rabbidva 3403	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝐴 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧})
70	65, 69	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧})
71	24	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
72		rabeq 3411	. . . . . . . . . . . . . . . 16 ⊢ (dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴 → {𝑤 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧})
73	71, 72	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → {𝑤 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧} = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧})
74	70, 73	eqtr4d 2767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} = {𝑤 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧})
75	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → ( ≤ ∩ (𝐴 × 𝐴)) ∈ V)
76	65, 71	eleqtrrd 2831	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → 𝑧 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)))
77	15	ordtopn1 23097	. . . . . . . . . . . . . . 15 ⊢ ((( ≤ ∩ (𝐴 × 𝐴)) ∈ V ∧ 𝑧 ∈ dom ( ≤ ∩ (𝐴 × 𝐴))) → {𝑤 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
78	75, 76, 77	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → {𝑤 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
79	74, 78	eqeltrd 2828	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
80	79	anassrs 467	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦)) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
81	80	expr 456	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → (𝑧 ≤ 𝑦 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
82	57, 81	syld 47	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
83	82	rexlimdva 3130	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
84	50, 83	biimtrid 242	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ≠ ∅ → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
85	45, 84	pm2.61dne 3011	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
86	85	rexlimdvaa 3131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝑥 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
87	37, 86	biimtrid 242	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
88	33, 87	pm2.61d 179	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
89	8, 88	eqeltrd 2828	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
90	89	ralrimiva 3121	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
91		fvex 6839	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
92	22, 91	eqeltri 2824	. . . . . 6 ⊢ 𝐵 ∈ V
93	92	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
94		rabexg 5279	. . . . 5 ⊢ (𝐵 ∈ V → {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∈ V)
95	93, 94	syl 17	. . . 4 ⊢ (𝜑 → {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∈ V)
96	95	ralrimivw 3125	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∈ V)
97		eqid 2729	. . . 4 ⊢ (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) = (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})
98		ineq1 4166	. . . . 5 ⊢ (𝑣 = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} → (𝑣 ∩ 𝐴) = ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴))
99	98	eleq1d 2813	. . . 4 ⊢ (𝑣 = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
100	97, 99	ralrnmptw 7032	. . 3 ⊢ (∀𝑧 ∈ 𝐵 {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∈ V → (∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ∀𝑧 ∈ 𝐵 ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
101	96, 100	syl 17	. 2 ⊢ (𝜑 → (∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ∀𝑧 ∈ 𝐵 ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
102	90, 101	mpbird 257	1 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  {crab 3396  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286   class class class wbr 5095   ↦ cmpt 5176   × cxp 5621  dom cdm 5623  ran crn 5624  ‘cfv 6486  Basecbs 17138  lecple 17186  ordTopcordt 17421   Proset cproset 18216  Posetcpo 18231  Tosetctos 18338  Topctop 22796  TopOnctopon 22813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fi 9320  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-4 12211  df-5 12212  df-6 12213  df-7 12214  df-8 12215  df-9 12216  df-dec 12610  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-ple 17199  df-topgen 17365  df-ordt 17423  df-proset 18218  df-poset 18237  df-toset 18339  df-top 22797  df-topon 22814  df-bases 22849

This theorem is referenced by:  ordtrest2NEW  33892