Theorem ordtrest2NEWlem 34113

Description: Lemma for ordtrest2NEW 34114. (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 4252	. . . . 5 ⊢ ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) = {𝑤 ∈ (𝐵 ∩ 𝐴) ∣ ¬ 𝑤 ≤ 𝑧}
2		ordtrest2NEW.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		sseqin2 4159	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
4	2, 3	sylib 219	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∩ 𝐴) = 𝐴)
5	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐵 ∩ 𝐴) = 𝐴)
6		rabeq 3406	. . . . . 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 2787	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧})
9		ordtNEW.l	. . . . . . . . . . . . 13 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
10		fvex 6847	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) ∈ V
11	10	inex1 5252	. . . . . . . . . . . . 13 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
12	9, 11	eqeltri 2836	. . . . . . . . . . . 12 ⊢ ≤ ∈ V
13	12	inex1 5252	. . . . . . . . . . 11 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) ∈ V)
15		eqid 2740	. . . . . . . . . . 11 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
16	15	ordttopon 23183	. . . . . . . . . 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 18382	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
20		posprs 18280	. . . . . . . . . . . 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 34108	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
24	21, 2, 23	syl2anc 590	. . . . . . . . . 10 ⊢ (𝜑 → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
25	24	fveq2d 6838	. . . . . . . . 9 ⊢ (𝜑 → (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
26	17, 25	eleqtrd 2842	. . . . . . . 8 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
27		toponmax 22916	. . . . . . . 8 ⊢ ((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
29	28	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → 𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
30		rabid2 3425	. . . . . . 7 ⊢ (𝐴 = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
31		eleq1 2828	. . . . . . 7 ⊢ (𝐴 = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} → (𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
32	30, 31	sylbir 236	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 → (𝐴 ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
33	29, 32	syl5ibcom 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
34		dfrex2 3067	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 𝑤 ≤ 𝑧 ↔ ¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
35		breq1 5082	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ≤ 𝑧 ↔ 𝑥 ≤ 𝑧))
36	35	cbvrexvw 3219	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 𝑤 ≤ 𝑧 ↔ ∃𝑥 ∈ 𝐴 𝑥 ≤ 𝑧)
37	34, 36	bitr3i 278	. . . . . 6 ⊢ (¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 ↔ ∃𝑥 ∈ 𝐴 𝑥 ≤ 𝑧)
38		ordttop 23190	. . . . . . . . . . . . 13 ⊢ (( ≤ ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
39	14, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
40	39	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top)
41		0opn 22894	. . . . . . . . . . 11 ⊢ ((ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ Top → ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
43	42	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
44		eleq1 2828	. . . . . . . . 9 ⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} = ∅ → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ∅ ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
45	43, 44	syl5ibrcom 248	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} = ∅ → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
46		rabn0 4324	. . . . . . . . . 10 ⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
47		breq1 5082	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧))
48	47	notbid 319	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (¬ 𝑤 ≤ 𝑧 ↔ ¬ 𝑦 ≤ 𝑧))
49	48	cbvrexvw 3219	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑧)
50	46, 49	bitri 276	. . . . . . . . 9 ⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ≠ ∅ ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑧)
51	18	ad3antrrr 736	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝐾 ∈ Toset)
52	2	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → 𝐴 ⊆ 𝐵)
53	52	sselda 3922	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
54		simpllr 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐵)
55	22, 9	trleile 33057	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ Toset ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑦 ≤ 𝑧 ∨ 𝑧 ≤ 𝑦))
56	51, 53, 54, 55	syl3anc 1379	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑧 ∨ 𝑧 ≤ 𝑦))
57	56	ord 870	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑧 → 𝑧 ≤ 𝑦))
58		an4 662	. . . . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑧 ∈ 𝐵 ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴 ↔ ∀𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
61	59, 60	sylib 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝐵 ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
62	61	r19.21bi 3232	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
63	62	an32s 658	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦) → 𝑧 ∈ 𝐴))
64	63	impr 455	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦))) → 𝑧 ∈ 𝐴)
65	58, 64	sylan2b 600	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → 𝑧 ∈ 𝐴)
66		brinxp 5704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑤 ≤ 𝑧 ↔ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧))
67	66	ancoms 459	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑤 ≤ 𝑧 ↔ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧))
68	67	notbid 319	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 ≤ 𝑧 ↔ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧))
69	68	rabbidva 3398	. . . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
72		rabeq 3406	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → 𝑧 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)))
77	15	ordtopn1 23184	. . . . . . . . . . . . . . 15 ⊢ ((( ≤ ∩ (𝐴 × 𝐴)) ∈ V ∧ 𝑧 ∈ dom ( ≤ ∩ (𝐴 × 𝐴))) → {𝑤 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)) ∣ ¬ 𝑤( ≤ ∩ (𝐴 × 𝐴))𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
78	75, 76, 77	syl2anc 590	. . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
80	79	anassrs 468	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦)) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
81	80	expr 457	. . . . . . . . . . 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 3141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → (∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
84	50, 83	biimtrid 243	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ≠ ∅ → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
85	45, 84	pm2.61dne 3021	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧)) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
86	85	rexlimdvaa 3142	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝑥 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
87	37, 86	biimtrid 243	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧 → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
88	33, 87	pm2.61d 180	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
89	8, 88	eqeltrd 2840	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
90	89	ralrimiva 3132	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
91		fvex 6847	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
92	22, 91	eqeltri 2836	. . . . . 6 ⊢ 𝐵 ∈ V
93	92	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
94		rabexg 5272	. . . . 5 ⊢ (𝐵 ∈ V → {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∈ V)
95	93, 94	syl 17	. . . 4 ⊢ (𝜑 → {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∈ V)
96	95	ralrimivw 3136	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∈ V)
97		eqid 2740	. . . 4 ⊢ (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) = (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})
98		ineq1 4149	. . . . 5 ⊢ (𝑣 = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} → (𝑣 ∩ 𝐴) = ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴))
99	98	eleq1d 2825	. . . 4 ⊢ (𝑣 = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
100	97, 99	ralrnmptw 7042	. . 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 258	1 ⊢ (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})(𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ∃wrex 3064  {crab 3392  Vcvv 3432   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268   class class class wbr 5079   ↦ cmpt 5160   × cxp 5623  dom cdm 5625  ran crn 5626  ‘cfv 6492  Basecbs 17177  lecple 17225  ordTopcordt 17461   Proset cproset 18256  Posetcpo 18271  Tosetctos 18378  Topctop 22883  TopOnctopon 22900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fi 9321  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-2 12242  df-3 12243  df-4 12244  df-5 12245  df-6 12246  df-7 12247  df-8 12248  df-9 12249  df-dec 12643  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17178  df-ress 17199  df-ple 17238  df-topgen 17404  df-ordt 17463  df-proset 18258  df-poset 18277  df-toset 18379  df-top 22884  df-topon 22901  df-bases 22936

This theorem is referenced by:  ordtrest2NEW  34114