filnetlem3 - Mathbox for Jeff Hankins

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Theorem filnetlem3 36923

Description: Lemma for filnet 36925. (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 6057	. . . . . 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 36922	. . . . . . . 8 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻))
5	4	simpli 489	. . . . . . 7 ⊢ ( I ↾ 𝐻) ⊆ 𝐷
6		dmss 5895	. . . . . . 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 3987	. . . . 5 ⊢ 𝐻 ⊆ dom 𝐷
9		ssun1 4134	. . . . 5 ⊢ dom 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷)
10	8, 9	sstri 3949	. . . 4 ⊢ 𝐻 ⊆ (dom 𝐷 ∪ ran 𝐷)
11		dmrnssfld 5967	. . . 4 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷
12	10, 11	sstri 3949	. . 3 ⊢ 𝐻 ⊆ ∪ ∪ 𝐷
13	4	simpri 491	. . . . 5 ⊢ 𝐷 ⊆ (𝐻 × 𝐻)
14		uniss 4883	. . . . 5 ⊢ (𝐷 ⊆ (𝐻 × 𝐻) → ∪ 𝐷 ⊆ ∪ (𝐻 × 𝐻))
15		uniss 4883	. . . . 5 ⊢ (∪ 𝐷 ⊆ ∪ (𝐻 × 𝐻) → ∪ ∪ 𝐷 ⊆ ∪ ∪ (𝐻 × 𝐻))
16	13, 14, 15	mp2b 10	. . . 4 ⊢ ∪ ∪ 𝐷 ⊆ ∪ ∪ (𝐻 × 𝐻)
17		unixpss 5800	. . . . 5 ⊢ ∪ ∪ (𝐻 × 𝐻) ⊆ (𝐻 ∪ 𝐻)
18		unidm 4114	. . . . 5 ⊢ (𝐻 ∪ 𝐻) = 𝐻
19	17, 18	sseqtri 3988	. . . 4 ⊢ ∪ ∪ (𝐻 × 𝐻) ⊆ 𝐻
20	16, 19	sstri 3949	. . 3 ⊢ ∪ ∪ 𝐷 ⊆ 𝐻
21	12, 20	eqssi 3956	. 2 ⊢ 𝐻 = ∪ ∪ 𝐷
22		filelss 24024	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → 𝑛 ⊆ 𝑋)
23		xpss2 5684	. . . . . . . 8 ⊢ (𝑛 ⊆ 𝑋 → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
24	22, 23	syl 18	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
25	24	ralrimiva 3160	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋))
26		ss2iun 4978	. . . . . 6 ⊢ (∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ({𝑛} × 𝑋) → ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑋))
27	25, 26	syl 18	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑋))
28		iunxpconst 5737	. . . . 5 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑋) = (𝐹 × 𝑋)
29	27, 28	sseqtrdi 3980	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ (𝐹 × 𝑋))
30	2, 29	eqsstrid 3978	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
31	5	a1i 11	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ( I ↾ 𝐻) ⊆ 𝐷)
32	3	relopabiv 5810	. . . . 5 ⊢ Rel 𝐷
33	31, 32	jctil 529	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷))
34		simpl 488	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝐹 ∈ (Fil‘𝑋))
35	30	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝐻 ⊆ (𝐹 × 𝑋))
36		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑣 ∈ 𝐻)
37	35, 36	sseldd 3941	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑣 ∈ (𝐹 × 𝑋))
38		xp1st 8020	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝐹 × 𝑋) → (1^st ‘𝑣) ∈ 𝐹)
39	37, 38	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (1^st ‘𝑣) ∈ 𝐹)
40		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑧 ∈ 𝐻)
41	35, 40	sseldd 3941	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → 𝑧 ∈ (𝐹 × 𝑋))
42		xp1st 8020	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝐹 × 𝑋) → (1^st ‘𝑧) ∈ 𝐹)
43	41, 42	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → (1^st ‘𝑧) ∈ 𝐹)
44		filinn0 24032	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (1^st ‘𝑣) ∈ 𝐹 ∧ (1^st ‘𝑧) ∈ 𝐹) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ≠ ∅)
45	34, 39, 43, 44	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ≠ ∅)
46		n0 4310	. . . . . . . . 9 ⊢ (((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ≠ ∅ ↔ ∃𝑢 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
47	45, 46	sylib 221	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ∃𝑢 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
48	36	adantr 486	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑣 ∈ 𝐻)
49		filin 24026	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ 𝐹)
51	50	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ 𝐹)
52		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
53		id 23	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) → 𝑛 = ((1^st ‘𝑣) ∩ (1^st ‘𝑧)))
54	53	opeliunxp2 5827	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 2877	. . . . . . . . . 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 6898	. . . . . . . . . . . . . 14 ⊢ (1^st ‘𝑣) ∈ V
58	57	inex1 5289	. . . . . . . . . . . . 13 ⊢ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ∈ V
59		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑢 ∈ V
60	58, 59	op1st 7996	. . . . . . . . . . . 12 ⊢ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) = ((1^st ‘𝑣) ∩ (1^st ‘𝑧))
61		inss1 4192	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ⊆ (1^st ‘𝑣)
62	60, 61	eqsstri 3986	. . . . . . . . . . 11 ⊢ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) ⊆ (1^st ‘𝑣)
63		vex 3462	. . . . . . . . . . . 12 ⊢ 𝑣 ∈ V
64		opex 5448	. . . . . . . . . . . 12 ⊢ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ V
65	2, 3, 63, 64	filnetlem1 36921	. . . . . . . . . . 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 723	. . . . . . . . . 10 ⊢ (𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ↔ (𝑣 ∈ 𝐻 ∧ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ 𝐻))
67	48, 56, 66	sylanbrc 595	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → 𝑧 ∈ 𝐻)
69		inss2 4193	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑣) ∩ (1^st ‘𝑧)) ⊆ (1^st ‘𝑧)
70	60, 69	eqsstri 3986	. . . . . . . . . . 11 ⊢ (1^st ‘⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩) ⊆ (1^st ‘𝑧)
71		vex 3462	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
72	2, 3, 71, 64	filnetlem1 36921	. . . . . . . . . . 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 723	. . . . . . . . . 10 ⊢ (𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ↔ (𝑧 ∈ 𝐻 ∧ ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ ∈ 𝐻))
74	68, 56, 73	sylanbrc 595	. . . . . . . . 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 5116	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ → (𝑣𝐷𝑤 ↔ 𝑣𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩))
76		breq2 5116	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩ → (𝑧𝐷𝑤 ↔ 𝑧𝐷⟨((1^st ‘𝑣) ∩ (1^st ‘𝑧)), 𝑢⟩))
77	75, 76	anbi12d 644	. . . . . . . . . 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 3568	. . . . . . . . 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 596	. . . . . . . 8 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) ∧ 𝑢 ∈ ((1^st ‘𝑣) ∩ (1^st ‘𝑧))) → ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
80	47, 79	exlimddv 1968	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻)) → ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
81	80	ralrimivva 3211	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑣 ∈ 𝐻 ∀𝑧 ∈ 𝐻 ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
82		codir 6123	. . . . . 6 ⊢ ((𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷) ↔ ∀𝑣 ∈ 𝐻 ∀𝑧 ∈ 𝐻 ∃𝑤(𝑣𝐷𝑤 ∧ 𝑧𝐷𝑤))
83	81, 82	sylibr 237	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷))
84		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
85	2, 3, 63, 84	filnetlem1 36921	. . . . . . . . . . . 12 ⊢ (𝑣𝐷𝑤 ↔ ((𝑣 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻) ∧ (1^st ‘𝑤) ⊆ (1^st ‘𝑣)))
86	85	simplbi 502	. . . . . . . . . . 11 ⊢ (𝑣𝐷𝑤 → (𝑣 ∈ 𝐻 ∧ 𝑤 ∈ 𝐻))
87	86	simpld 500	. . . . . . . . . 10 ⊢ (𝑣𝐷𝑤 → 𝑣 ∈ 𝐻)
88	2, 3, 84, 71	filnetlem1 36921	. . . . . . . . . . . 12 ⊢ (𝑤𝐷𝑧 ↔ ((𝑤 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1^st ‘𝑧) ⊆ (1^st ‘𝑤)))
89	88	simplbi 502	. . . . . . . . . . 11 ⊢ (𝑤𝐷𝑧 → (𝑤 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻))
90	89	simprd 501	. . . . . . . . . 10 ⊢ (𝑤𝐷𝑧 → 𝑧 ∈ 𝐻)
91	87, 90	anim12i 625	. . . . . . . . 9 ⊢ ((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → (𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻))
92	88	simprbi 503	. . . . . . . . . 10 ⊢ (𝑤𝐷𝑧 → (1^st ‘𝑧) ⊆ (1^st ‘𝑤))
93	85	simprbi 503	. . . . . . . . . 10 ⊢ (𝑣𝐷𝑤 → (1^st ‘𝑤) ⊆ (1^st ‘𝑣))
94	92, 93	sylan9ssr 3954	. . . . . . . . 9 ⊢ ((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → (1^st ‘𝑧) ⊆ (1^st ‘𝑣))
95	2, 3, 63, 71	filnetlem1 36921	. . . . . . . . 9 ⊢ (𝑣𝐷𝑧 ↔ ((𝑣 ∈ 𝐻 ∧ 𝑧 ∈ 𝐻) ∧ (1^st ‘𝑧) ⊆ (1^st ‘𝑣)))
96	91, 94, 95	sylanbrc 595	. . . . . . . 8 ⊢ ((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧)
97	96	ax-gen 1828	. . . . . . 7 ⊢ ∀𝑧((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧)
98	97	gen2 1829	. . . . . 6 ⊢ ∀𝑣∀𝑤∀𝑧((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧)
99		cotr 6115	. . . . . 6 ⊢ ((𝐷 ∘ 𝐷) ⊆ 𝐷 ↔ ∀𝑣∀𝑤∀𝑧((𝑣𝐷𝑤 ∧ 𝑤𝐷𝑧) → 𝑣𝐷𝑧))
100	98, 99	mpbir 234	. . . . 5 ⊢ (𝐷 ∘ 𝐷) ⊆ 𝐷
101	83, 100	jctil 529	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝐷 ∘ 𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷)))
102		filtop 24027	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
103		xpexg 7751	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋 ∈ 𝐹) → (𝐹 × 𝑋) ∈ V)
104	102, 103	mpdan 700	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
105	104, 30	ssexd 5298	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
106	105, 105	xpexd 7752	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 × 𝐻) ∈ V)
107		ssexg 5293	. . . . . 6 ⊢ ((𝐷 ⊆ (𝐻 × 𝐻) ∧ (𝐻 × 𝐻) ∈ V) → 𝐷 ∈ V)
108	13, 106, 107	sylancr 599	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ V)
109	21	isdir 18664	. . . . 5 ⊢ (𝐷 ∈ V → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷 ∘ 𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷)))))
110	108, 109	syl 18	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐷 ∈ DirRel ↔ ((Rel 𝐷 ∧ ( I ↾ 𝐻) ⊆ 𝐷) ∧ ((𝐷 ∘ 𝐷) ⊆ 𝐷 ∧ (𝐻 × 𝐻) ⊆ (^◡𝐷 ∘ 𝐷)))))
111	33, 101, 110	mpbir2and 726	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
112	30, 111	jca 521	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
113	21, 112	pm3.2i 476	1 ⊢ (𝐻 = ∪ ∪ 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1568 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 Vcvv 3458 ∪ cun 3906 ∩ cin 3907 ⊆ wss 3908 ∅c0 4289 {csn 4592 ⟨cop 4598 ∪ cuni 4875 ∪ ciun 4959 class class class wbr 5112 {copab 5176 I cid 5558 × cxp 5662 ^◡ccnv 5663 dom cdm 5664 ran crn 5665 ↾ cres 5666 ∘ ccom 5668 Rel wrel 5669 ‘cfv 6540 1^st c1st 7986 DirRelcdir 18660 Filcfil 24017

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-iun 4961 df-br 5113 df-opab 5177 df-mpt 5196 df-id 5559 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1st 7988 df-dir 18662 df-fbas 21534 df-fil 24018

This theorem is used by: filnetlem4 36924

W3C validator