Theorem fnemeet2 32699

Description: The meet of equivalence classes under the fineness relation-part two. (Contributed by Jeff Hankins, 6-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
fnemeet2	⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)))

Distinct variable groups:   𝑦,𝑡,𝑥,𝑆   𝑡,𝑉,𝑥   𝑡,𝑋,𝑥,𝑦   𝑡,𝑇,𝑥

Allowed substitution hints:   𝑇(𝑦)   𝑉(𝑦)

Proof of Theorem fnemeet2

Step	Hyp	Ref	Expression
1		riin0 4728	. . . . . . . . . 10 ⊢ (𝑆 = ∅ → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = 𝒫 𝑋)
2	1	unieqd 4584	. . . . . . . . 9 ⊢ (𝑆 = ∅ → ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∪ 𝒫 𝑋)
3		unipw 5046	. . . . . . . . 9 ⊢ ∪ 𝒫 𝑋 = 𝑋
4	2, 3	syl6req 2822	. . . . . . . 8 ⊢ (𝑆 = ∅ → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
5	4	a1i 11	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 = ∅ → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
6		n0 4078	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
7		unieq 4582	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
8	7	eqeq2d 2781	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑥))
9	8	rspccva 3459	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
10	9	3adant1 1124	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
11		fnemeet1 32698	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝑥)
12		eqid 2771	. . . . . . . . . . . . 13 ⊢ ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
13		eqid 2771	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 = ∪ 𝑥
14	12, 13	fnebas 32676	. . . . . . . . . . . 12 ⊢ ((𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝑥 → ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∪ 𝑥)
15	11, 14	syl 17	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∪ 𝑥)
16	10, 15	eqtr4d 2808	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
17	16	3expia 1114	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑥 ∈ 𝑆 → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
18	17	exlimdv 2013	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (∃𝑥 𝑥 ∈ 𝑆 → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
19	6, 18	syl5bi 232	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 ≠ ∅ → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
20	5, 19	pm2.61dne 3029	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
21	20	adantr 466	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
22		eqid 2771	. . . . . . 7 ⊢ ∪ 𝑇 = ∪ 𝑇
23	22, 12	fnebas 32676	. . . . . 6 ⊢ (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → ∪ 𝑇 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
24	23	adantl 467	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))) → ∪ 𝑇 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
25	21, 24	eqtr4d 2808	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))) → 𝑋 = ∪ 𝑇)
26	25	ex 397	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → 𝑋 = ∪ 𝑇))
27		fnetr 32683	. . . . . . 7 ⊢ ((𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∧ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝑥) → 𝑇Fne𝑥)
28	27	expcom 398	. . . . . 6 ⊢ ((𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝑥 → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → 𝑇Fne𝑥))
29	11, 28	syl 17	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → 𝑇Fne𝑥))
30	29	3expa 1111	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑥 ∈ 𝑆) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → 𝑇Fne𝑥))
31	30	ralrimdva 3118	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → ∀𝑥 ∈ 𝑆 𝑇Fne𝑥))
32	26, 31	jcad 496	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)))
33		simprl 746	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑋 = ∪ 𝑇)
34	20	adantr 466	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
35	33, 34	eqtr3d 2807	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → ∪ 𝑇 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
36		eqimss2 3807	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝑇 → ∪ 𝑇 ⊆ 𝑋)
37	36	ad2antrl 699	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → ∪ 𝑇 ⊆ 𝑋)
38		sspwuni 4745	. . . . . . 7 ⊢ (𝑇 ⊆ 𝒫 𝑋 ↔ ∪ 𝑇 ⊆ 𝑋)
39	37, 38	sylibr 224	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ 𝒫 𝑋)
40		breq2 4790	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (𝑇Fne𝑥 ↔ 𝑇Fne𝑡))
41	40	cbvralv 3320	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑆 𝑇Fne𝑥 ↔ ∀𝑡 ∈ 𝑆 𝑇Fne𝑡)
42		fnetg 32677	. . . . . . . . . 10 ⊢ (𝑇Fne𝑡 → 𝑇 ⊆ (topGen‘𝑡))
43	42	ralimi 3101	. . . . . . . . 9 ⊢ (∀𝑡 ∈ 𝑆 𝑇Fne𝑡 → ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
44	41, 43	sylbi 207	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑆 𝑇Fne𝑥 → ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
45	44	ad2antll 700	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
46		ssiin 4704	. . . . . . 7 ⊢ (𝑇 ⊆ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ↔ ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
47	45, 46	sylibr 224	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
48	39, 47	ssind 3985	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
49		pwexg 4981	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
50		inex1g 4935	. . . . . . . 8 ⊢ (𝒫 𝑋 ∈ V → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∈ V)
51	49, 50	syl 17	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∈ V)
52	51	ad2antrr 697	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∈ V)
53		bastg 20991	. . . . . 6 ⊢ ((𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∈ V → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ⊆ (topGen‘(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
54	52, 53	syl 17	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ⊆ (topGen‘(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
55	48, 54	sstrd 3762	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ (topGen‘(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
56	22, 12	isfne4 32672	. . . 4 ⊢ (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ↔ (∪ 𝑇 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∧ 𝑇 ⊆ (topGen‘(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))))
57	35, 55, 56	sylanbrc 564	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
58	57	ex 397	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → ((𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥) → 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
59	32, 58	impbid 202	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631  ∃wex 1852   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  ∅c0 4063  𝒫 cpw 4297  ∪ cuni 4574  ∩ ciin 4655   class class class wbr 4786  ‘cfv 6031  topGenctg 16306  Fnecfne 32668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-topgen 16312  df-fne 32669

This theorem is referenced by: (None)