Theorem fnemeet2 36333

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 5105	. . . . . . . . . 10 ⊢ (𝑆 = ∅ → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = 𝒫 𝑋)
2	1	unieqd 4944	. . . . . . . . 9 ⊢ (𝑆 = ∅ → ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∪ 𝒫 𝑋)
3		unipw 5470	. . . . . . . . 9 ⊢ ∪ 𝒫 𝑋 = 𝑋
4	2, 3	eqtr2di 2797	. . . . . . . 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 4376	. . . . . . . 8 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑆)
7		unieq 4942	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → ∪ 𝑦 = ∪ 𝑥)
8	7	eqeq2d 2751	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑥))
9	8	rspccva 3634	. . . . . . . . . . . 12 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
10	9	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ 𝑥)
11		fnemeet1 36332	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝑥)
12		eqid 2740	. . . . . . . . . . . . 13 ⊢ ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
13		eqid 2740	. . . . . . . . . . . . 13 ⊢ ∪ 𝑥 = ∪ 𝑥
14	12, 13	fnebas 36310	. . . . . . . . . . . 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 2783	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑥 ∈ 𝑆) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
17	16	3expia 1121	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑥 ∈ 𝑆 → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
18	17	exlimdv 1932	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (∃𝑥 𝑥 ∈ 𝑆 → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
19	6, 18	biimtrid 242	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑆 ≠ ∅ → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
20	5, 19	pm2.61dne 3034	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
21	20	adantr 480	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
22		eqid 2740	. . . . . . 7 ⊢ ∪ 𝑇 = ∪ 𝑇
23	22, 12	fnebas 36310	. . . . . 6 ⊢ (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → ∪ 𝑇 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
24	23	adantl 481	. . . . 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 2783	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))) → 𝑋 = ∪ 𝑇)
26	25	ex 412	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → 𝑋 = ∪ 𝑇))
27		fnetr 36317	. . . . . . 7 ⊢ ((𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∧ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝑥) → 𝑇Fne𝑥)
28	27	expcom 413	. . . . . 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 1118	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ 𝑥 ∈ 𝑆) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → 𝑇Fne𝑥))
31	30	ralrimdva 3160	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → ∀𝑥 ∈ 𝑆 𝑇Fne𝑥))
32	26, 31	jcad 512	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) → (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)))
33		simprl 770	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑋 = ∪ 𝑇)
34	20	adantr 480	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑋 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
35	33, 34	eqtr3d 2782	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → ∪ 𝑇 = ∪ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
36		eqimss2 4068	. . . . . . . 8 ⊢ (𝑋 = ∪ 𝑇 → ∪ 𝑇 ⊆ 𝑋)
37	36	ad2antrl 727	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → ∪ 𝑇 ⊆ 𝑋)
38		sspwuni 5123	. . . . . . 7 ⊢ (𝑇 ⊆ 𝒫 𝑋 ↔ ∪ 𝑇 ⊆ 𝑋)
39	37, 38	sylibr 234	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ 𝒫 𝑋)
40		breq2 5170	. . . . . . . . . 10 ⊢ (𝑥 = 𝑡 → (𝑇Fne𝑥 ↔ 𝑇Fne𝑡))
41	40	cbvralvw 3243	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑆 𝑇Fne𝑥 ↔ ∀𝑡 ∈ 𝑆 𝑇Fne𝑡)
42		fnetg 36311	. . . . . . . . . 10 ⊢ (𝑇Fne𝑡 → 𝑇 ⊆ (topGen‘𝑡))
43	42	ralimi 3089	. . . . . . . . 9 ⊢ (∀𝑡 ∈ 𝑆 𝑇Fne𝑡 → ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
44	41, 43	sylbi 217	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑆 𝑇Fne𝑥 → ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
45	44	ad2antll 728	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
46		ssiin 5078	. . . . . . 7 ⊢ (𝑇 ⊆ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ↔ ∀𝑡 ∈ 𝑆 𝑇 ⊆ (topGen‘𝑡))
47	45, 46	sylibr 234	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
48	39, 47	ssind 4262	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
49		pwexg 5396	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
50		inex1g 5337	. . . . . . . 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 725	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ∈ V)
53		bastg 22994	. . . . . 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 4019	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇 ⊆ (topGen‘(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
56	22, 12	isfne4 36306	. . . 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 582	. . 3 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) ∧ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)) → 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)))
58	57	ex 412	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → ((𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥) → 𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))))
59	32, 58	impbid 212	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦) → (𝑇Fne(𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) ↔ (𝑋 = ∪ 𝑇 ∧ ∀𝑥 ∈ 𝑆 𝑇Fne𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931  ∩ ciin 5016   class class class wbr 5166  ‘cfv 6573  topGenctg 17497  Fnecfne 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-topgen 17503  df-fne 36303

This theorem is referenced by: (None)