filnetlem3 - Mathbox for Jeff Hankins

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Theorem filnetlem3 34569

Description: Lemma for filnet 34571. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)

**Hypotheses**
Ref	Expression
filnet.h	⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
filnet.d	⊢ 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))}

**Assertion**
Ref	Expression
filnetlem3	⊢ (𝐻 = ∪ ∪ 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))

Distinct variable groups: 𝑥,𝑦,𝑛,𝐹 𝑥,𝐻,𝑦 𝑛,𝑋 Allowed substitution hints: 𝐷(𝑥,𝑦,𝑛) 𝐻(𝑛) 𝑋(𝑥,𝑦)

Proof of Theorem filnetlem3 Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmresi 5961	. . . . . 6 ⊢ dom ( I ↾ 𝐻) = 𝐻
2		filnet.h	. . . . . . . . 9 ⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
3		filnet.d	. . . . . . . . 9 ⊢ 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1^st ‘𝑦) ⊆ (1^st ‘𝑥))}
4	2, 3	filnetlem2 34568	. . . . . . . 8 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻))
5	4	simpli 484	. . . . . . 7 ⊢ ( I ↾ 𝐻) ⊆ 𝐷
6		dmss 5811	. . . . . . 7 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
7	5, 6	ax-mp 5	. . . . . 6 ⊢ dom ( I ↾ 𝐻) ⊆ dom 𝐷
8	1, 7	eqsstrri 3956	. . . . 5 ⊢ 𝐻 ⊆ dom 𝐷
9		ssun1 4106	. . . . 5 ⊢ dom 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
10	8, 9	sstri 3930	. . . 4 ⊢ 𝐻 ⊆ (dom 𝐷 ∪ ran 𝐷)
11		dmrnssfld 5879	. . . 4 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷
12	10, 11	sstri 3930	. . 3 ⊢ 𝐻 ⊆ ∪ ∪ 𝐷
13	4	simpri 486	. . . . 5 ⊢ 𝐷 ⊆ (𝐻 × 𝐻)
14		uniss 4847	. . . . 5 ⊢ (𝐷 ⊆ (𝐻 × 𝐻) → ∪ 𝐷 ⊆ ∪ (𝐻 × 𝐻))
15		uniss 4847	. . . . 5 ⊢ (∪ 𝐷 ⊆ ∪ (𝐻 × 𝐻) → ∪ ∪ 𝐷 ⊆ ∪ ∪ (𝐻 × 𝐻))
16	13, 14, 15	mp2b 10	. . . 4 ⊢ ∪ ∪ 𝐷 ⊆ ∪ ∪ (𝐻 × 𝐻)
17		unixpss 5720	. . . . 5 ⊢ ∪ ∪ (𝐻 × 𝐻) ⊆ (𝐻 ∪ 𝐻)
18		unidm 4086	. . . . 5 ⊢ (𝐻 ∪ 𝐻) = 𝐻
19	17, 18	sseqtri 3957	. . . 4 ⊢ ∪ ∪ (𝐻 × 𝐻) ⊆ 𝐻
20	16, 19	sstri 3930	. . 3 ⊢ ∪ ∪ 𝐷 ⊆ 𝐻
21	12, 20	eqssi 3937	. 2 ⊢ 𝐻 = ∪ ∪ 𝐷
22		filelss 23003	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → 𝑛 ⊆ 𝑋)
23		xpss2 5609	. . . . . . . 8 ⊢ (𝑛 ⊆ 𝑋 → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
25	24	ralrimiva 3103	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
26		ss2iun 4942	. . . . . 6 ⊢ (∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋) → ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑋))
27	25, 26	syl 17	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑋))
28		iunxpconst 5659	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑋) = (𝐹 × 𝑋)
29	27, 28	sseqtrdi 3971	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ (𝐹 × 𝑋))
30	2, 29	eqsstrid 3969	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
31	5	a1i 11	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ( I ↾ 𝐻) ⊆ 𝐷)
32	3	relopabiv 5730	. . . . 5 ⊢ Rel 𝐷
33	31, 32	jctil 520	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷))
34		simpl 483	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝐹 ∈ (Fil‘𝑋))
35	30	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝐻 ⊆ (𝐹 × 𝑋))
36		simprl 768	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑣 ∈ 𝐻)
37	35, 36	sseldd 3922	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑣 ∈ (𝐹 × 𝑋))
38		xp1st 7863	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝐹 × 𝑋) → (1^st ‘𝑣) ∈ 𝐹)
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (1^st ‘𝑣) ∈ 𝐹)
40		simprr 770	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑧 ∈ 𝐻)
41	35, 40	sseldd 3922	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑧 ∈ (𝐹 × 𝑋))
42		xp1st 7863	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐹 × 𝑋) → (1^st ‘𝑧) ∈ 𝐹)
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (1^st ‘𝑧) ∈ 𝐹)
44		filinn0 23011	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (1^st ‘𝑣) ∈ 𝐹 ∧ (1^st ‘𝑧) ∈ 𝐹) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ≠ ∅)
45	34, 39, 43, 44	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ≠ ∅)
46		n0 4280	. . . . . . . . 9 ⊢ (((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
47	45, 46	sylib 217	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ∃𝑢 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
48	36	adantr 481	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑣 ∈ 𝐻)
49		filin 23005	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (1^st ‘𝑣) ∈ 𝐹 ∧ (1^st ‘𝑧) ∈ 𝐹) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ 𝐹)
50	34, 39, 43, 49	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ 𝐹)
51	50	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ 𝐹)
52		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
53		id 22	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) → 𝑛 = ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
54	53	opeliunxp2 5747	. . . . . . . . . . . 12 ⊢ (⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ↔ (((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ 𝐹 ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))))
55	51, 52, 54	sylanbrc 583	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛))
56	55, 2	eleqtrrdi 2850	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ 𝐻)
57		fvex 6787	. . . . . . . . . . . . . 14 ⊢ (1^st ‘𝑣) ∈ V
58	57	inex1 5241	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ V
59		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
60	58, 59	op1st 7839	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) = ((1^st ‘𝑣) ∩ (1^st ‘𝑧))
61		inss1 4162	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ⊆ (1^st ‘𝑣)
62	60, 61	eqsstri 3955	. . . . . . . . . . 11 ⊢ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) ⊆ (1^st ‘𝑣)
63		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
64		opex 5379	. . . . . . . . . . . 12 ⊢ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ V
65	2, 3, 63, 64	filnetlem1 34567	. . . . . . . . . . 11 ⊢ (𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ↔ ((𝑣 ∈ 𝐻 ∧ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) ⊆ (1^st ‘𝑣)))
66	62, 65	mpbiran2 707	. . . . . . . . . 10 ⊢ (𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ↔ (𝑣 ∈ 𝐻 ∧ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ 𝐻))
67	48, 56, 66	sylanbrc 583	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩)
68	40	adantr 481	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑧 ∈ 𝐻)
69		inss2 4163	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ⊆ (1^st ‘𝑧)
70	60, 69	eqsstri 3955	. . . . . . . . . . 11 ⊢ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) ⊆ (1^st ‘𝑧)
71		vex 3436	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
72	2, 3, 71, 64	filnetlem1 34567	. . . . . . . . . . 11 ⊢ (𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ↔ ((𝑧 ∈ 𝐻 ∧ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ 𝐻) ∧ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) ⊆ (1^st ‘𝑧)))
73	70, 72	mpbiran2 707	. . . . . . . . . 10 ⊢ (𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ↔ (𝑧 ∈ 𝐻 ∧ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ 𝐻))
74	68, 56, 73	sylanbrc 583	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩)
75		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ → (𝑣𝐷𝑤 ↔ 𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩))
76		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ → (𝑧𝐷𝑤 ↔ 𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩))
77	75, 76	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑤 = ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ → ((𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤) ↔ (𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩)))
78	64, 77	spcev 3545	. . . . . . . . 9 ⊢ ((𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∧ 𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) → ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
79	67, 74, 78	syl2anc 584	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
80	47, 79	exlimddv 1938	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
81	80	ralrimivva 3123	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑣 ∈ 𝐻 ∀𝑧 ∈ 𝐻 ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
82		codir 6025	. . . . . 6 ⊢ ((𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷) ↔ ∀𝑣 ∈ 𝐻 ∀𝑧 ∈ 𝐻 ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
83	81, 82	sylibr 233	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷))
84		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
85	2, 3, 63, 84	filnetlem1 34567	. . . . . . . . . . . 12 ⊢ (𝑣𝐷𝑤 ↔ ((𝑣 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻) ∧ (1^st ‘𝑤) ⊆ (1^st ‘𝑣)))
86	85	simplbi 498	. . . . . . . . . . 11 ⊢ (𝑣𝐷𝑤 → (𝑣 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻))
87	86	simpld 495	. . . . . . . . . 10 ⊢ (𝑣𝐷𝑤 → 𝑣 ∈ 𝐻)
88	2, 3, 84, 71	filnetlem1 34567	. . . . . . . . . . . 12 ⊢ (𝑤𝐷𝑧 ↔ ((𝑤 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1^st ‘𝑧) ⊆ (1^st ‘𝑤)))
89	88	simplbi 498	. . . . . . . . . . 11 ⊢ (𝑤𝐷𝑧 → (𝑤 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻))
90	89	simprd 496	. . . . . . . . . 10 ⊢ (𝑤𝐷𝑧 → 𝑧 ∈ 𝐻)
91	87, 90	anim12i 613	. . . . . . . . 9 ⊢ ((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻))
92	88	simprbi 497	. . . . . . . . . 10 ⊢ (𝑤𝐷𝑧 → (1^st ‘𝑧) ⊆ (1^st ‘𝑤))
93	85	simprbi 497	. . . . . . . . . 10 ⊢ (𝑣𝐷𝑤 → (1^st ‘𝑤) ⊆ (1^st ‘𝑣))
94	92, 93	sylan9ssr 3935	. . . . . . . . 9 ⊢ ((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → (1^st ‘𝑧) ⊆ (1^st ‘𝑣))
95	2, 3, 63, 71	filnetlem1 34567	. . . . . . . . 9 ⊢ (𝑣𝐷𝑧 ↔ ((𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1^st ‘𝑧) ⊆ (1^st ‘𝑣)))
96	91, 94, 95	sylanbrc 583	. . . . . . . 8 ⊢ ((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧)
97	96	ax-gen 1798	. . . . . . 7 ⊢ ∀𝑧((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧)
98	97	gen2 1799	. . . . . 6 ⊢ ∀𝑣∀𝑤∀𝑧((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧)
99		cotr 6017	. . . . . 6 ⊢ ((𝐷 ∘ 𝐷) ⊆ 𝐷 ↔ ∀𝑣∀𝑤∀𝑧((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧))
100	98, 99	mpbir 230	. . . . 5 ⊢ (𝐷 ∘ 𝐷) ⊆ 𝐷
101	83, 100	jctil 520	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝐷 ∘ 𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷)))
102		filtop 23006	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
103		xpexg 7600	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋 ∈ 𝐹) → (𝐹 × 𝑋) ∈ V)
104	102, 103	mpdan 684	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
105	104, 30	ssexd 5248	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
106	105, 105	xpexd 7601	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ∈ V)
107		ssexg 5247	. . . . . 6 ⊢ ((𝐷 ⊆ (𝐻 × 𝐻) ∧ (𝐻 × 𝐻) ∈ V) → 𝐷 ∈ V)
108	13, 106, 107	sylancr 587	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ V)
109	21	isdir 18316	. . . . 5 ⊢ (𝐷 ∈ V → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷 ∘ 𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷)))))
110	108, 109	syl 17	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷 ∘ 𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷)))))
111	33, 101, 110	mpbir2and 710	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
112	30, 111	jca 512	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
113	21, 112	pm3.2i 471	1 ⊢ (𝐻 = ∪ ∪ 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 Vcvv 3432 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 {csn 4561 ⟨cop 4567 ∪ cuni 4839 ∪ ciun 4924 class class class wbr 5074 {copab 5136 I cid 5488 × cxp 5587 ^◡ccnv 5588 dom cdm 5589 ran crn 5590 ↾ cres 5591 ∘ ccom 5593 Rel wrel 5594 ‘cfv 6433 1^st c1st 7829 DirRelcdir 18312 Filcfil 22996

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-1st 7831 df-dir 18314 df-fbas 20594 df-fil 22997

This theorem is referenced by: filnetlem4 34570

W3C validator