Theorem filnetlem4 31546

Description: Lemma for filnet 31547. (Contributed by Jeff Hankins, 15-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) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))}

Assertion

Ref	Expression
filnetlem4	⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))

Distinct variable groups:   𝑥,𝑦   𝑓,𝑑,𝑛,𝑥,𝑦,𝐹   𝐻,𝑑,𝑓,𝑥,𝑦   𝐷,𝑑,𝑓   𝑋,𝑑,𝑓,𝑛

Allowed substitution hints:   𝐷(𝑥,𝑦,𝑛)   𝐻(𝑛)   𝑋(𝑥,𝑦)

Proof of Theorem filnetlem4

Dummy variables 𝑘 𝑚 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		filnet.h	. . . . 5 ⊢ 𝐻 = ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)
2		filnet.d	. . . . 5 ⊢ 𝐷 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ (1st ‘𝑦) ⊆ (1st ‘𝑥))}
3	1, 2	filnetlem3 31545	. . . 4 ⊢ (𝐻 = ∪ ∪ 𝐷 ∧ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel)))
4	3	simpri 477	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐻 ⊆ (𝐹 × 𝑋) ∧ 𝐷 ∈ DirRel))
5	4	simprd 478	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐷 ∈ DirRel)
6		f2ndres 7082	. . . . 5 ⊢ (2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋
7	4	simpld 474	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ⊆ (𝐹 × 𝑋))
8		fssres2 5985	. . . . 5 ⊢ (((2nd ↾ (𝐹 × 𝑋)):(𝐹 × 𝑋)⟶𝑋 ∧ 𝐻 ⊆ (𝐹 × 𝑋)) → (2nd ↾ 𝐻):𝐻⟶𝑋)
9	6, 7, 8	sylancr 694	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (2nd ↾ 𝐻):𝐻⟶𝑋)
10		filtop 21469	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐹)
11		xpexg 6858	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑋 ∈ 𝐹) → (𝐹 × 𝑋) ∈ V)
12	10, 11	mpdan 699	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝐹 × 𝑋) ∈ V)
13	12, 7	ssexd 4733	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ∈ V)
14		fex 6394	. . . 4 ⊢ (((2nd ↾ 𝐻):𝐻⟶𝑋 ∧ 𝐻 ∈ V) → (2nd ↾ 𝐻) ∈ V)
15	9, 13, 14	syl2anc 691	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (2nd ↾ 𝐻) ∈ V)
16		dirdm 17057	. . . . . . . 8 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷)
17	5, 16	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → dom 𝐷 = ∪ ∪ 𝐷)
18	3	simpli 473	. . . . . . 7 ⊢ 𝐻 = ∪ ∪ 𝐷
19	17, 18	syl6reqr 2663	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 = dom 𝐷)
20	19	feq2d 5944	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((2nd ↾ 𝐻):𝐻⟶𝑋 ↔ (2nd ↾ 𝐻):dom 𝐷⟶𝑋))
21	9, 20	mpbid 221	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (2nd ↾ 𝐻):dom 𝐷⟶𝑋)
22		eqid 2610	. . . . . . . . . . . . . 14 ⊢ dom 𝐷 = dom 𝐷
23	22	tailf 31540	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷)
24	5, 23	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷)
25	19	feq2d 5944	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 ↔ (tail‘𝐷):dom 𝐷⟶𝒫 dom 𝐷))
26	24, 25	mpbird 246	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
27	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (tail‘𝐷):𝐻⟶𝒫 dom 𝐷)
28		ffn 5958	. . . . . . . . . 10 ⊢ ((tail‘𝐷):𝐻⟶𝒫 dom 𝐷 → (tail‘𝐷) Fn 𝐻)
29		imaeq2 5381	. . . . . . . . . . . 12 ⊢ (𝑑 = ((tail‘𝐷)‘𝑓) → ((2nd ↾ 𝐻) “ 𝑑) = ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)))
30	29	sseq1d 3595	. . . . . . . . . . 11 ⊢ (𝑑 = ((tail‘𝐷)‘𝑓) → (((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
31	30	rexrn 6269	. . . . . . . . . 10 ⊢ ((tail‘𝐷) Fn 𝐻 → (∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓 ∈ 𝐻 ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
32	27, 28, 31	3syl 18	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑓 ∈ 𝐻 ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡))
33		fo2nd 7080	. . . . . . . . . . . . . . 15 ⊢ 2nd :V–onto→V
34		fofn 6030	. . . . . . . . . . . . . . 15 ⊢ (2nd :V–onto→V → 2nd Fn V)
35	33, 34	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 2nd Fn V
36		ssv 3588	. . . . . . . . . . . . . 14 ⊢ 𝐻 ⊆ V
37		fnssres 5918	. . . . . . . . . . . . . 14 ⊢ ((2nd Fn V ∧ 𝐻 ⊆ V) → (2nd ↾ 𝐻) Fn 𝐻)
38	35, 36, 37	mp2an 704	. . . . . . . . . . . . 13 ⊢ (2nd ↾ 𝐻) Fn 𝐻
39		fnfun 5902	. . . . . . . . . . . . 13 ⊢ ((2nd ↾ 𝐻) Fn 𝐻 → Fun (2nd ↾ 𝐻))
40	38, 39	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun (2nd ↾ 𝐻)
41	27	ffvelrnda 6267	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ∈ 𝒫 dom 𝐷)
42	41	elpwid 4118	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom 𝐷)
43	19	ad2antrr 758	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝐻 = dom 𝐷)
44	42, 43	sseqtr4d 3605	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ⊆ 𝐻)
45		fndm 5904	. . . . . . . . . . . . . 14 ⊢ ((2nd ↾ 𝐻) Fn 𝐻 → dom (2nd ↾ 𝐻) = 𝐻)
46	38, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ dom (2nd ↾ 𝐻) = 𝐻
47	44, 46	syl6sseqr 3615	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((tail‘𝐷)‘𝑓) ⊆ dom (2nd ↾ 𝐻))
48		funimass4 6157	. . . . . . . . . . . 12 ⊢ ((Fun (2nd ↾ 𝐻) ∧ ((tail‘𝐷)‘𝑓) ⊆ dom (2nd ↾ 𝐻)) → (((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡))
49	40, 47, 48	sylancr 694	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡))
50	5	ad2antrr 758	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝐷 ∈ DirRel)
51		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ 𝐻)
52	51, 43	eleqtrd 2690	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ dom 𝐷)
53		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑑 ∈ V
54	53	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → 𝑑 ∈ V)
55	22	eltail 31539	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ DirRel ∧ 𝑓 ∈ dom 𝐷 ∧ 𝑑 ∈ V) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
56	50, 52, 54, 55	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ 𝑓𝐷𝑑))
57	51	biantrurd 528	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (𝑑 ∈ 𝐻 ↔ (𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻)))
58	57	anbi1d 737	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) ↔ ((𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻) ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓))))
59		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑓 ∈ V
60	1, 2, 59, 53	filnetlem1 31543	. . . . . . . . . . . . . . . 16 ⊢ (𝑓𝐷𝑑 ↔ ((𝑓 ∈ 𝐻 ∧ 𝑑 ∈ 𝐻) ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)))
61	58, 60	syl6bbr 277	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) ↔ 𝑓𝐷𝑑))
62	56, 61	bitr4d 270	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (𝑑 ∈ ((tail‘𝐷)‘𝑓) ↔ (𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓))))
63	62	imbi1d 330	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) → ((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡)))
64		fvres 6117	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ 𝐻 → ((2nd ↾ 𝐻)‘𝑑) = (2nd ‘𝑑))
65	64	eleq1d 2672	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ 𝐻 → (((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd ‘𝑑) ∈ 𝑡))
66	65	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) → (((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡 ↔ (2nd ‘𝑑) ∈ 𝑡))
67	66	pm5.74i 259	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) → ((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ ((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) → (2nd ‘𝑑) ∈ 𝑡))
68		impexp 461	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) → (2nd ‘𝑑) ∈ 𝑡) ↔ (𝑑 ∈ 𝐻 → ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡)))
69	67, 68	bitri 263	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ 𝐻 ∧ (1st ‘𝑑) ⊆ (1st ‘𝑓)) → ((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑 ∈ 𝐻 → ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡)))
70	63, 69	syl6bb 275	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → ((𝑑 ∈ ((tail‘𝐷)‘𝑓) → ((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡) ↔ (𝑑 ∈ 𝐻 → ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡))))
71	70	ralbidv2 2967	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (∀𝑑 ∈ ((tail‘𝐷)‘𝑓)((2nd ↾ 𝐻)‘𝑑) ∈ 𝑡 ↔ ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡)))
72	49, 71	bitrd 267	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑓 ∈ 𝐻) → (((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡)))
73	72	rexbidva 3031	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑓 ∈ 𝐻 ((2nd ↾ 𝐻) “ ((tail‘𝐷)‘𝑓)) ⊆ 𝑡 ↔ ∃𝑓 ∈ 𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡)))
74		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑘 ∈ V
75		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ V
76	74, 75	op1std 7069	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ⟨𝑘, 𝑣⟩ → (1st ‘𝑑) = 𝑘)
77	76	sseq1d 3595	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = ⟨𝑘, 𝑣⟩ → ((1st ‘𝑑) ⊆ (1st ‘𝑓) ↔ 𝑘 ⊆ (1st ‘𝑓)))
78	74, 75	op2ndd 7070	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = ⟨𝑘, 𝑣⟩ → (2nd ‘𝑑) = 𝑣)
79	78	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = ⟨𝑘, 𝑣⟩ → ((2nd ‘𝑑) ∈ 𝑡 ↔ 𝑣 ∈ 𝑡))
80	77, 79	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑑 = ⟨𝑘, 𝑣⟩ → (((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡) ↔ (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡)))
81	80	raliunxp 5183	. . . . . . . . . . . . 13 ⊢ (∀𝑑 ∈ ∪ 𝑘 ∈ 𝐹 ({𝑘} × 𝑘)((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡) ↔ ∀𝑘 ∈ 𝐹 ∀𝑣 ∈ 𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡))
82		sneq 4135	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → {𝑛} = {𝑘})
83		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑘 → 𝑛 = 𝑘)
84	82, 83	xpeq12d 5064	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑘 → ({𝑛} × 𝑛) = ({𝑘} × 𝑘))
85	84	cbviunv 4495	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) = ∪ 𝑘 ∈ 𝐹 ({𝑘} × 𝑘)
86	1, 85	eqtri 2632	. . . . . . . . . . . . . 14 ⊢ 𝐻 = ∪ 𝑘 ∈ 𝐹 ({𝑘} × 𝑘)
87	86	raleqi 3119	. . . . . . . . . . . . 13 ⊢ (∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡) ↔ ∀𝑑 ∈ ∪ 𝑘 ∈ 𝐹 ({𝑘} × 𝑘)((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡))
88		dfss3 3558	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ⊆ 𝑡 ↔ ∀𝑣 ∈ 𝑘 𝑣 ∈ 𝑡)
89	88	imbi2i 325	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ (𝑘 ⊆ (1st ‘𝑓) → ∀𝑣 ∈ 𝑘 𝑣 ∈ 𝑡))
90		r19.21v 2943	. . . . . . . . . . . . . . 15 ⊢ (∀𝑣 ∈ 𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡) ↔ (𝑘 ⊆ (1st ‘𝑓) → ∀𝑣 ∈ 𝑘 𝑣 ∈ 𝑡))
91	89, 90	bitr4i 266	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∀𝑣 ∈ 𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡))
92	91	ralbii 2963	. . . . . . . . . . . . 13 ⊢ (∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∀𝑘 ∈ 𝐹 ∀𝑣 ∈ 𝑘 (𝑘 ⊆ (1st ‘𝑓) → 𝑣 ∈ 𝑡))
93	81, 87, 92	3bitr4i 291	. . . . . . . . . . . 12 ⊢ (∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡) ↔ ∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡))
94	93	rexbii 3023	. . . . . . . . . . 11 ⊢ (∃𝑓 ∈ 𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡) ↔ ∃𝑓 ∈ 𝐻 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡))
95	1	rexeqi 3120	. . . . . . . . . . 11 ⊢ (∃𝑓 ∈ 𝐻 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∃𝑓 ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡))
96		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑛 ∈ V
97		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑚 ∈ V
98	96, 97	op1std 7069	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ⟨𝑛, 𝑚⟩ → (1st ‘𝑓) = 𝑛)
99	98	sseq2d 3596	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ⟨𝑛, 𝑚⟩ → (𝑘 ⊆ (1st ‘𝑓) ↔ 𝑘 ⊆ 𝑛))
100	99	imbi1d 330	. . . . . . . . . . . . 13 ⊢ (𝑓 = ⟨𝑛, 𝑚⟩ → ((𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)))
101	100	ralbidv 2969	. . . . . . . . . . . 12 ⊢ (𝑓 = ⟨𝑛, 𝑚⟩ → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)))
102	101	rexiunxp 5184	. . . . . . . . . . 11 ⊢ (∃𝑓 ∈ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛)∀𝑘 ∈ 𝐹 (𝑘 ⊆ (1st ‘𝑓) → 𝑘 ⊆ 𝑡) ↔ ∃𝑛 ∈ 𝐹 ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))
103	94, 95, 102	3bitri 285	. . . . . . . . . 10 ⊢ (∃𝑓 ∈ 𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡) ↔ ∃𝑛 ∈ 𝐹 ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))
104		fileln0 21464	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → 𝑛 ≠ ∅)
105	104	adantlr 747	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → 𝑛 ≠ ∅)
106		r19.9rzv 4017	. . . . . . . . . . . . 13 ⊢ (𝑛 ≠ ∅ → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)))
107	105, 106	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡)))
108		ssid 3587	. . . . . . . . . . . . . . 15 ⊢ 𝑛 ⊆ 𝑛
109		sseq1 3589	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑘 ⊆ 𝑛 ↔ 𝑛 ⊆ 𝑛))
110		sseq1 3589	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑘 ⊆ 𝑡 ↔ 𝑛 ⊆ 𝑡))
111	109, 110	imbi12d 333	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ((𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ (𝑛 ⊆ 𝑛 → 𝑛 ⊆ 𝑡)))
112	111	rspcv 3278	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝐹 → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) → (𝑛 ⊆ 𝑛 → 𝑛 ⊆ 𝑡)))
113	108, 112	mpii 45	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐹 → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) → 𝑛 ⊆ 𝑡))
114	113	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) → 𝑛 ⊆ 𝑡))
115		sstr2 3575	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ⊆ 𝑛 → (𝑛 ⊆ 𝑡 → 𝑘 ⊆ 𝑡))
116	115	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑛 ⊆ 𝑡 → (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))
117	116	ralrimivw 2950	. . . . . . . . . . . . 13 ⊢ (𝑛 ⊆ 𝑡 → ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡))
118	114, 117	impbid1 214	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ 𝑛 ⊆ 𝑡))
119	107, 118	bitr3d 269	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) ∧ 𝑛 ∈ 𝐹) → (∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ 𝑛 ⊆ 𝑡))
120	119	rexbidva 3031	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑛 ∈ 𝐹 ∃𝑚 ∈ 𝑛 ∀𝑘 ∈ 𝐹 (𝑘 ⊆ 𝑛 → 𝑘 ⊆ 𝑡) ↔ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡))
121	103, 120	syl5bb 271	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑓 ∈ 𝐻 ∀𝑑 ∈ 𝐻 ((1st ‘𝑑) ⊆ (1st ‘𝑓) → (2nd ‘𝑑) ∈ 𝑡) ↔ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡))
122	32, 73, 121	3bitrd 293	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑡 ⊆ 𝑋) → (∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡 ↔ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡))
123	122	pm5.32da 671	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡)))
124		filn0 21476	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅)
125	96	snnz 4252	. . . . . . . . . . . . . . . 16 ⊢ {𝑛} ≠ ∅
126	104, 125	jctil 558	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅))
127		neanior 2874	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} ≠ ∅ ∧ 𝑛 ≠ ∅) ↔ ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
128	126, 127	sylib 207	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ¬ ({𝑛} = ∅ ∨ 𝑛 = ∅))
129		ss0b 3925	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} × 𝑛) = ∅)
130		xpeq0 5473	. . . . . . . . . . . . . . 15 ⊢ (({𝑛} × 𝑛) = ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
131	129, 130	bitri 263	. . . . . . . . . . . . . 14 ⊢ (({𝑛} × 𝑛) ⊆ ∅ ↔ ({𝑛} = ∅ ∨ 𝑛 = ∅))
132	128, 131	sylnibr 318	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑛 ∈ 𝐹) → ¬ ({𝑛} × 𝑛) ⊆ ∅)
133	132	ralrimiva 2949	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∀𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
134		r19.2z 4012	. . . . . . . . . . . 12 ⊢ ((𝐹 ≠ ∅ ∧ ∀𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅) → ∃𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
135	124, 133, 134	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅)
136		rexnal 2978	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ 𝐹 ¬ ({𝑛} × 𝑛) ⊆ ∅ ↔ ¬ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅)
137	135, 136	sylib 207	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘𝑋) → ¬ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅)
138	1	sseq1i 3592	. . . . . . . . . . . 12 ⊢ (𝐻 ⊆ ∅ ↔ ∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅)
139		ss0b 3925	. . . . . . . . . . . 12 ⊢ (𝐻 ⊆ ∅ ↔ 𝐻 = ∅)
140		iunss 4497	. . . . . . . . . . . 12 ⊢ (∪ 𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅ ↔ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅)
141	138, 139, 140	3bitr3i 289	. . . . . . . . . . 11 ⊢ (𝐻 = ∅ ↔ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅)
142	141	necon3abii 2828	. . . . . . . . . 10 ⊢ (𝐻 ≠ ∅ ↔ ¬ ∀𝑛 ∈ 𝐹 ({𝑛} × 𝑛) ⊆ ∅)
143	137, 142	sylibr 223	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐻 ≠ ∅)
144		dmresi 5376	. . . . . . . . . . . 12 ⊢ dom ( I ↾ 𝐻) = 𝐻
145	1, 2	filnetlem2 31544	. . . . . . . . . . . . . 14 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 ∧ 𝐷 ⊆ (𝐻 × 𝐻))
146	145	simpli 473	. . . . . . . . . . . . 13 ⊢ ( I ↾ 𝐻) ⊆ 𝐷
147		dmss 5245	. . . . . . . . . . . . 13 ⊢ (( I ↾ 𝐻) ⊆ 𝐷 → dom ( I ↾ 𝐻) ⊆ dom 𝐷)
148	146, 147	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom ( I ↾ 𝐻) ⊆ dom 𝐷
149	144, 148	eqsstr3i 3599	. . . . . . . . . . 11 ⊢ 𝐻 ⊆ dom 𝐷
150	145	simpri 477	. . . . . . . . . . . . 13 ⊢ 𝐷 ⊆ (𝐻 × 𝐻)
151		dmss 5245	. . . . . . . . . . . . 13 ⊢ (𝐷 ⊆ (𝐻 × 𝐻) → dom 𝐷 ⊆ dom (𝐻 × 𝐻))
152	150, 151	ax-mp 5	. . . . . . . . . . . 12 ⊢ dom 𝐷 ⊆ dom (𝐻 × 𝐻)
153		dmxpid 5266	. . . . . . . . . . . 12 ⊢ dom (𝐻 × 𝐻) = 𝐻
154	152, 153	sseqtri 3600	. . . . . . . . . . 11 ⊢ dom 𝐷 ⊆ 𝐻
155	149, 154	eqssi 3584	. . . . . . . . . 10 ⊢ 𝐻 = dom 𝐷
156	155	tailfb 31542	. . . . . . . . 9 ⊢ ((𝐷 ∈ DirRel ∧ 𝐻 ≠ ∅) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
157	5, 143, 156	syl2anc 691	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → ran (tail‘𝐷) ∈ (fBas‘𝐻))
158		elfm 21561	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐹 ∧ ran (tail‘𝐷) ∈ (fBas‘𝐻) ∧ (2nd ↾ 𝐻):𝐻⟶𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡)))
159	10, 157, 9, 158	syl3anc 1318	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑑 ∈ ran (tail‘𝐷)((2nd ↾ 𝐻) “ 𝑑) ⊆ 𝑡)))
160		filfbas 21462	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
161		elfg 21485	. . . . . . . 8 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡)))
162	160, 161	syl 17	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑛 ∈ 𝐹 𝑛 ⊆ 𝑡)))
163	123, 159, 162	3bitr4d 299	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) ↔ 𝑡 ∈ (𝑋filGen𝐹)))
164	163	eqrdv 2608	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)) = (𝑋filGen𝐹))
165		fgfil 21489	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
166	164, 165	eqtr2d 2645	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)))
167	21, 166	jca 553	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((2nd ↾ 𝐻):dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷))))
168		feq1 5939	. . . . 5 ⊢ (𝑓 = (2nd ↾ 𝐻) → (𝑓:dom 𝐷⟶𝑋 ↔ (2nd ↾ 𝐻):dom 𝐷⟶𝑋))
169		oveq2 6557	. . . . . . 7 ⊢ (𝑓 = (2nd ↾ 𝐻) → (𝑋 FilMap 𝑓) = (𝑋 FilMap (2nd ↾ 𝐻)))
170	169	fveq1d 6105	. . . . . 6 ⊢ (𝑓 = (2nd ↾ 𝐻) → ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)))
171	170	eqeq2d 2620	. . . . 5 ⊢ (𝑓 = (2nd ↾ 𝐻) → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)) ↔ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷))))
172	168, 171	anbi12d 743	. . . 4 ⊢ (𝑓 = (2nd ↾ 𝐻) → ((𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))) ↔ ((2nd ↾ 𝐻):dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷)))))
173	172	spcegv 3267	. . 3 ⊢ ((2nd ↾ 𝐻) ∈ V → (((2nd ↾ 𝐻):dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap (2nd ↾ 𝐻))‘ran (tail‘𝐷))) → ∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
174	15, 167, 173	sylc 63	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))
175		dmeq 5246	. . . . . 6 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
176	175	feq2d 5944	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝑓:dom 𝑑⟶𝑋 ↔ 𝑓:dom 𝐷⟶𝑋))
177		fveq2 6103	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (tail‘𝑑) = (tail‘𝐷))
178	177	rneqd 5274	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ran (tail‘𝑑) = ran (tail‘𝐷))
179	178	fveq2d 6107	. . . . . 6 ⊢ (𝑑 = 𝐷 → ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))
180	179	eqeq2d 2620	. . . . 5 ⊢ (𝑑 = 𝐷 → (𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑)) ↔ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷))))
181	176, 180	anbi12d 743	. . . 4 ⊢ (𝑑 = 𝐷 → ((𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ (𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
182	181	exbidv 1837	. . 3 ⊢ (𝑑 = 𝐷 → (∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))) ↔ ∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))))
183	182	rspcev 3282	. 2 ⊢ ((𝐷 ∈ DirRel ∧ ∃𝑓(𝑓:dom 𝐷⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝐷)))) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))
184	5, 174, 183	syl2anc 691	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ∃𝑑 ∈ DirRel ∃𝑓(𝑓:dom 𝑑⟶𝑋 ∧ 𝐹 = ((𝑋 FilMap 𝑓)‘ran (tail‘𝑑))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583  {copab 4642   I cid 4948   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  DirRelcdir 17051  tailctail 17052  fBascfbas 19555  filGencfg 19556  Filcfil 21459   FilMap cfm 21547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-dir 17053  df-tail 17054  df-fbas 19564  df-fg 19565  df-fil 21460  df-fm 21552

This theorem is referenced by:  filnet  31547