Theorem fnemeet1 33714

Description: The meet of a collection of equivalence classes of covers with respect to fineness. (Contributed by Jeff Hankins, 5-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
fnemeet1	⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝐴)

Distinct variable groups:   𝑦,𝑡,𝐴   𝑡,𝑆,𝑦   𝑡,𝑉   𝑡,𝑋,𝑦

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem fnemeet1

Step	Hyp	Ref	Expression
1		unitg 21574	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑆 → ∪ (topGen‘𝑡) = ∪ 𝑡)
2	1	adantl 484	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ (topGen‘𝑡) = ∪ 𝑡)
3		unieq 4848	. . . . . . . . . 10 ⊢ (𝑦 = 𝑡 → ∪ 𝑦 = ∪ 𝑡)
4	3	eqeq2d 2832	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑡))
5	4	rspccva 3621	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝑡)
6	5	3ad2antl2 1182	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝑡)
7	2, 6	eqtr4d 2859	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ (topGen‘𝑡) = 𝑋)
8		eqimss 4022	. . . . . 6 ⊢ (∪ (topGen‘𝑡) = 𝑋 → ∪ (topGen‘𝑡) ⊆ 𝑋)
9	7, 8	syl 17	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ (topGen‘𝑡) ⊆ 𝑋)
10		sspwuni 5021	. . . . 5 ⊢ ((topGen‘𝑡) ⊆ 𝒫 𝑋 ↔ ∪ (topGen‘𝑡) ⊆ 𝑋)
11	9, 10	sylibr 236	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → (topGen‘𝑡) ⊆ 𝒫 𝑋)
12	11	ralrimiva 3182	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ 𝒫 𝑋)
13		ne0i 4299	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
14	13	3ad2ant3 1131	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
15		riinn0 5004	. . 3 ⊢ ((∀𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
16	12, 14, 15	syl2anc 586	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
17		simp3 1134	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
18		ssid 3988	. . . . . . . 8 ⊢ (topGen‘𝐴) ⊆ (topGen‘𝐴)
19		fveq2 6669	. . . . . . . . . 10 ⊢ (𝑡 = 𝐴 → (topGen‘𝑡) = (topGen‘𝐴))
20	19	sseq1d 3997	. . . . . . . . 9 ⊢ (𝑡 = 𝐴 → ((topGen‘𝑡) ⊆ (topGen‘𝐴) ↔ (topGen‘𝐴) ⊆ (topGen‘𝐴)))
21	20	rspcev 3622	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐴)) → ∃𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
22	17, 18, 21	sylancl 588	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∃𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
23		iinss 4979	. . . . . . 7 ⊢ (∃𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴) → ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
24	22, 23	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
25	24	unissd 4847	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ ∪ (topGen‘𝐴))
26		unitg 21574	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → ∪ (topGen‘𝐴) = ∪ 𝐴)
27	26	3ad2ant3 1131	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ (topGen‘𝐴) = ∪ 𝐴)
28	25, 27	sseqtrd 4006	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ ∪ 𝐴)
29		unieq 4848	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
30	29	eqeq2d 2832	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
31	30	rspccva 3621	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
32	31	3adant1 1126	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
33	32	adantr 483	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
34	33, 6	eqtr3d 2858	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝐴 = ∪ 𝑡)
35		simpr 487	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ 𝑆)
36		ssid 3988	. . . . . . . . 9 ⊢ 𝑡 ⊆ 𝑡
37		eltg3i 21568	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑆 ∧ 𝑡 ⊆ 𝑡) → ∪ 𝑡 ∈ (topGen‘𝑡))
38	35, 36, 37	sylancl 588	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝑡 ∈ (topGen‘𝑡))
39	34, 38	eqeltrd 2913	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝐴 ∈ (topGen‘𝑡))
40	39	ralrimiva 3182	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑡 ∈ 𝑆 ∪ 𝐴 ∈ (topGen‘𝑡))
41		uniexg 7465	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 → ∪ 𝐴 ∈ V)
42	41	3ad2ant3 1131	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ∈ V)
43		eliin 4923	. . . . . . 7 ⊢ (∪ 𝐴 ∈ V → (∪ 𝐴 ∈ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ↔ ∀𝑡 ∈ 𝑆 ∪ 𝐴 ∈ (topGen‘𝑡)))
44	42, 43	syl 17	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (∪ 𝐴 ∈ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ↔ ∀𝑡 ∈ 𝑆 ∪ 𝐴 ∈ (topGen‘𝑡)))
45	40, 44	mpbird 259	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ∈ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
46		elssuni 4867	. . . . 5 ⊢ (∪ 𝐴 ∈ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) → ∪ 𝐴 ⊆ ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
47	45, 46	syl 17	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ⊆ ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
48	28, 47	eqssd 3983	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ 𝐴)
49		eqid 2821	. . . 4 ⊢ ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)
50		eqid 2821	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
51	49, 50	isfne4 33688	. . 3 ⊢ (∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)Fne𝐴 ↔ (∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ 𝐴 ∧ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴)))
52	48, 24, 51	sylanbrc 585	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)Fne𝐴)
53	16, 52	eqbrtrd 5087	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  ∪ cuni 4837  ∩ ciin 4919   class class class wbr 5065  ‘cfv 6354  topGenctg 16710  Fnecfne 33684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iin 4921  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-topgen 16716  df-fne 33685

This theorem is referenced by:  fnemeet2  33715