Theorem ordtrest2NEWlem 33953

Description: Lemma for ordtrest2NEW 33954. (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 4292	. . . . 5 ⊢ ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) = {𝑤 ∈ (𝐵 ∩ 𝐴) ∣ ¬ 𝑤 ≤ 𝑧}
2		ordtrest2NEW.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		sseqin2 4198	. . . . . . . 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 3430	. . . . . 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 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧})
9		ordtNEW.l	. . . . . . . . . . . . 13 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵))
10		fvex 6889	. . . . . . . . . . . . . 14 ⊢ (le‘𝐾) ∈ V
11	10	inex1 5287	. . . . . . . . . . . . 13 ⊢ ((le‘𝐾) ∩ (𝐵 × 𝐵)) ∈ V
12	9, 11	eqeltri 2830	. . . . . . . . . . . 12 ⊢ ≤ ∈ V
13	12	inex1 5287	. . . . . . . . . . 11 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) ∈ V
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ( ≤ ∩ (𝐴 × 𝐴)) ∈ V)
15		eqid 2735	. . . . . . . . . . 11 ⊢ dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ( ≤ ∩ (𝐴 × 𝐴))
16	15	ordttopon 23131	. . . . . . . . . 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 18430	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
20		posprs 18328	. . . . . . . . . . . 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 33948	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
24	21, 2, 23	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴)
25	24	fveq2d 6880	. . . . . . . . 9 ⊢ (𝜑 → (TopOn‘dom ( ≤ ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
26	17, 25	eleqtrd 2836	. . . . . . . 8 ⊢ (𝜑 → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
27		toponmax 22864	. . . . . . . 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 3449	. . . . . . 7 ⊢ (𝐴 = {𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
31		eleq1 2822	. . . . . . 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 3063	. . . . . . 7 ⊢ (∃𝑤 ∈ 𝐴 𝑤 ≤ 𝑧 ↔ ¬ ∀𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
35		breq1 5122	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝑤 ≤ 𝑧 ↔ 𝑥 ≤ 𝑧))
36	35	cbvrexvw 3221	. . . . . . 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 23138	. . . . . . . . . . . . 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 22842	. . . . . . . . . . 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 2822	. . . . . . . . 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 4364	. . . . . . . . . 10 ⊢ ({𝑤 ∈ 𝐴 ∣ ¬ 𝑤 ≤ 𝑧} ≠ ∅ ↔ ∃𝑤 ∈ 𝐴 ¬ 𝑤 ≤ 𝑧)
47		breq1 5122	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑦 → (𝑤 ≤ 𝑧 ↔ 𝑦 ≤ 𝑧))
48	47	notbid 318	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (¬ 𝑤 ≤ 𝑧 ↔ ¬ 𝑦 ≤ 𝑧))
49	48	cbvrexvw 3221	. . . . . . . . . 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 3958	. . . . . . . . . . . . 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 32951	. . . . . . . . . . . . 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 4047	. . . . . . . . . . . . . . . . . . . . 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 3234	. . . . . . . . . . . . . . . . . . 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 5733	. . . . . . . . . . . . . . . . . . 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 3422	. . . . . . . . . . . . . . . 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 3430	. . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐵) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ≤ 𝑧) ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ≤ 𝑦))) → 𝑧 ∈ dom ( ≤ ∩ (𝐴 × 𝐴)))
77	15	ordtopn1 23132	. . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . 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 3141	. . . . . . . . 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 3018	. . . . . . 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 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 2834	. . 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 6889	. . . . . . 7 ⊢ (Base‘𝐾) ∈ V
92	22, 91	eqeltri 2830	. . . . . 6 ⊢ 𝐵 ∈ V
93	92	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
94		rabexg 5307	. . . . 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 2735	. . . 4 ⊢ (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧}) = (𝑧 ∈ 𝐵 ↦ {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧})
98		ineq1 4188	. . . . 5 ⊢ (𝑣 = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} → (𝑣 ∩ 𝐴) = ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴))
99	98	eleq1d 2819	. . . 4 ⊢ (𝑣 = {𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} → ((𝑣 ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) ↔ ({𝑤 ∈ 𝐵 ∣ ¬ 𝑤 ≤ 𝑧} ∩ 𝐴) ∈ (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))))
100	97, 99	ralrnmptw 7084	. . 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 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3415  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308   class class class wbr 5119   ↦ cmpt 5201   × cxp 5652  dom cdm 5654  ran crn 5655  ‘cfv 6531  Basecbs 17228  lecple 17278  ordTopcordt 17513   Proset cproset 18304  Posetcpo 18319  Tosetctos 18426  Topctop 22831  TopOnctopon 22848

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fi 9423  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-dec 12709  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-ple 17291  df-topgen 17457  df-ordt 17515  df-proset 18306  df-poset 18325  df-toset 18427  df-top 22832  df-topon 22849  df-bases 22884

This theorem is referenced by:  ordtrest2NEW  33954