Theorem fnemeet1 36509

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 22909	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑆 → ∪ (topGen‘𝑡) = ∪ 𝑡)
2	1	adantl 481	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ (topGen‘𝑡) = ∪ 𝑡)
3		unieq 4872	. . . . . . . . . 10 ⊢ (𝑦 = 𝑡 → ∪ 𝑦 = ∪ 𝑡)
4	3	eqeq2d 2745	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑡))
5	4	rspccva 3573	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝑡)
6	5	3ad2antl2 1187	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝑡)
7	2, 6	eqtr4d 2772	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ (topGen‘𝑡) = 𝑋)
8		eqimss 3990	. . . . . 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 5053	. . . . 5 ⊢ ((topGen‘𝑡) ⊆ 𝒫 𝑋 ↔ ∪ (topGen‘𝑡) ⊆ 𝑋)
11	9, 10	sylibr 234	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → (topGen‘𝑡) ⊆ 𝒫 𝑋)
12	11	ralrimiva 3126	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ 𝒫 𝑋)
13		ne0i 4291	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
14	13	3ad2ant3 1135	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
15		riinn0 5036	. . 3 ⊢ ((∀𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
16	12, 14, 15	syl2anc 584	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
17		simp3 1138	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
18		ssid 3954	. . . . . . . 8 ⊢ (topGen‘𝐴) ⊆ (topGen‘𝐴)
19		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑡 = 𝐴 → (topGen‘𝑡) = (topGen‘𝐴))
20	19	sseq1d 3963	. . . . . . . . 9 ⊢ (𝑡 = 𝐴 → ((topGen‘𝑡) ⊆ (topGen‘𝐴) ↔ (topGen‘𝐴) ⊆ (topGen‘𝐴)))
21	20	rspcev 3574	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐴)) → ∃𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
22	17, 18, 21	sylancl 586	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∃𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
23		iinss 5010	. . . . . . 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 4871	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ ∪ (topGen‘𝐴))
26		unitg 22909	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → ∪ (topGen‘𝐴) = ∪ 𝐴)
27	26	3ad2ant3 1135	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ (topGen‘𝐴) = ∪ 𝐴)
28	25, 27	sseqtrd 3968	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ ∪ 𝐴)
29		unieq 4872	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
30	29	eqeq2d 2745	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
31	30	rspccva 3573	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
32	31	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
33	32	adantr 480	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
34	33, 6	eqtr3d 2771	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝐴 = ∪ 𝑡)
35		simpr 484	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ 𝑆)
36		ssid 3954	. . . . . . . . 9 ⊢ 𝑡 ⊆ 𝑡
37		eltg3i 22903	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑆 ∧ 𝑡 ⊆ 𝑡) → ∪ 𝑡 ∈ (topGen‘𝑡))
38	35, 36, 37	sylancl 586	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝑡 ∈ (topGen‘𝑡))
39	34, 38	eqeltrd 2834	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝐴 ∈ (topGen‘𝑡))
40	39	ralrimiva 3126	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑡 ∈ 𝑆 ∪ 𝐴 ∈ (topGen‘𝑡))
41		uniexg 7683	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 → ∪ 𝐴 ∈ V)
42	41	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ∈ V)
43		eliin 4949	. . . . . . 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 257	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ∈ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
46		elssuni 4892	. . . . 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 3949	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ 𝐴)
49		eqid 2734	. . . 4 ⊢ ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)
50		eqid 2734	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
51	49, 50	isfne4 36483	. . 3 ⊢ (∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)Fne𝐴 ↔ (∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ 𝐴 ∧ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴)))
52	48, 24, 51	sylanbrc 583	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)Fne𝐴)
53	16, 52	eqbrtrd 5118	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  ∪ cuni 4861  ∩ ciin 4945   class class class wbr 5096  ‘cfv 6490  topGenctg 17355  Fnecfne 36479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-topgen 17361  df-fne 36480

This theorem is referenced by:  fnemeet2  36510