Theorem fnemeet1 32361

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 20771	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑆 → ∪ (topGen‘𝑡) = ∪ 𝑡)
2	1	adantl 482	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ (topGen‘𝑡) = ∪ 𝑡)
3		unieq 4444	. . . . . . . . . 10 ⊢ (𝑦 = 𝑡 → ∪ 𝑦 = ∪ 𝑡)
4	3	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑡))
5	4	rspccva 3308	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝑡)
6	5	3ad2antl2 1224	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝑡)
7	2, 6	eqtr4d 2659	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ (topGen‘𝑡) = 𝑋)
8		eqimss 3657	. . . . . 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 4611	. . . . 5 ⊢ ((topGen‘𝑡) ⊆ 𝒫 𝑋 ↔ ∪ (topGen‘𝑡) ⊆ 𝑋)
11	9, 10	sylibr 224	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → (topGen‘𝑡) ⊆ 𝒫 𝑋)
12	11	ralrimiva 2966	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ 𝒫 𝑋)
13		ne0i 3921	. . . 4 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
14	13	3ad2ant3 1084	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑆 ≠ ∅)
15		riinn0 4595	. . 3 ⊢ ((∀𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
16	12, 14, 15	syl2anc 693	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)) = ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
17		simp3 1063	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝑆)
18		ssid 3624	. . . . . . . 8 ⊢ (topGen‘𝐴) ⊆ (topGen‘𝐴)
19		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑡 = 𝐴 → (topGen‘𝑡) = (topGen‘𝐴))
20	19	sseq1d 3632	. . . . . . . . 9 ⊢ (𝑡 = 𝐴 → ((topGen‘𝑡) ⊆ (topGen‘𝐴) ↔ (topGen‘𝐴) ⊆ (topGen‘𝐴)))
21	20	rspcev 3309	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ (topGen‘𝐴) ⊆ (topGen‘𝐴)) → ∃𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
22	17, 18, 21	sylancl 694	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∃𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴))
23		iinss 4571	. . . . . . 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 4462	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ ∪ (topGen‘𝐴))
26		unitg 20771	. . . . . 6 ⊢ (𝐴 ∈ 𝑆 → ∪ (topGen‘𝐴) = ∪ 𝐴)
27	26	3ad2ant3 1084	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ (topGen‘𝐴) = ∪ 𝐴)
28	25, 27	sseqtrd 3641	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ ∪ 𝐴)
29		unieq 4444	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
30	29	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝐴))
31	30	rspccva 3308	. . . . . . . . . . 11 ⊢ ((∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
32	31	3adant1 1079	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
33	32	adantr 481	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑋 = ∪ 𝐴)
34	33, 6	eqtr3d 2658	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝐴 = ∪ 𝑡)
35		simpr 477	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ 𝑆)
36		ssid 3624	. . . . . . . . 9 ⊢ 𝑡 ⊆ 𝑡
37		eltg3i 20765	. . . . . . . . 9 ⊢ ((𝑡 ∈ 𝑆 ∧ 𝑡 ⊆ 𝑡) → ∪ 𝑡 ∈ (topGen‘𝑡))
38	35, 36, 37	sylancl 694	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝑡 ∈ (topGen‘𝑡))
39	34, 38	eqeltrd 2701	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → ∪ 𝐴 ∈ (topGen‘𝑡))
40	39	ralrimiva 2966	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∀𝑡 ∈ 𝑆 ∪ 𝐴 ∈ (topGen‘𝑡))
41		uniexg 6955	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑆 → ∪ 𝐴 ∈ V)
42	41	3ad2ant3 1084	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ∈ V)
43		eliin 4525	. . . . . . 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 247	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ 𝐴 ∈ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))
46		elssuni 4467	. . . . 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 3620	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ 𝐴)
49		eqid 2622	. . . 4 ⊢ ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)
50		eqid 2622	. . . 4 ⊢ ∪ 𝐴 = ∪ 𝐴
51	49, 50	isfne4 32335	. . 3 ⊢ (∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)Fne𝐴 ↔ (∪ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) = ∪ 𝐴 ∧ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡) ⊆ (topGen‘𝐴)))
52	48, 24, 51	sylanbrc 698	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡)Fne𝐴)
53	16, 52	eqbrtrd 4675	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝑆 𝑋 = ∪ 𝑦 ∧ 𝐴 ∈ 𝑆) → (𝒫 𝑋 ∩ ∩ 𝑡 ∈ 𝑆 (topGen‘𝑡))Fne𝐴)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ∩ ciin 4521   class class class wbr 4653  ‘cfv 5888  topGenctg 16098  Fnecfne 32331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-topgen 16104  df-fne 32332

This theorem is referenced by:  fnemeet2  32362