Theorem alexsubb 24395

Description: Biconditional form of the Alexander Subbase Theorem alexsub 24394. (Contributed by Mario Carneiro, 27-Aug-2015.)

Assertion

Ref	Expression
alexsubb	⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑋,𝑦

Proof of Theorem alexsubb

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ ∪ (topGen‘(fi‘𝐵)) = ∪ (topGen‘(fi‘𝐵))
2	1	iscmp 23736	. . . 4 ⊢ ((topGen‘(fi‘𝐵)) ∈ Comp ↔ ((topGen‘(fi‘𝐵)) ∈ Top ∧ ∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦)))
3	2	simprbi 503	. . 3 ⊢ ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
4		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ 𝐵)
5		elex 3475	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ UFL → 𝑋 ∈ V)
6	5	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 ∈ V)
7	4, 6	eqeltrrd 2865	. . . . . . . . . 10 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 ∈ V)
8		uniexb 7802	. . . . . . . . . 10 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
9	7, 8	sylibr 237	. . . . . . . . 9 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ∈ V)
10		fiuni 9447	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
11	9, 10	syl 18	. . . . . . . 8 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 = ∪ (fi‘𝐵))
12		fibas 23325	. . . . . . . . 9 ⊢ (fi‘𝐵) ∈ TopBases
13		unitg 23315	. . . . . . . . 9 ⊢ ((fi‘𝐵) ∈ TopBases → ∪ (topGen‘(fi‘𝐵)) = ∪ (fi‘𝐵))
14	12, 13	mp1i 14	. . . . . . . 8 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘(fi‘𝐵)) = ∪ (fi‘𝐵))
15	11, 4, 14	3eqtr4d 2808	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ (topGen‘(fi‘𝐵)))
16	15	eqeq1d 2765	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑥 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥))
17	15	eqeq1d 2765	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑦 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
18	17	rexbidv 3190	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
19	16, 18	imbi12d 347	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) ↔ (∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦)))
20	19	ralbidv 3189	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) ↔ ∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦)))
21		ssfii 9438	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
22	9, 21	syl 18	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23		bastg 23314	. . . . . . . 8 ⊢ ((fi‘𝐵) ∈ TopBases → (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵)))
24	12, 23	ax-mp 5	. . . . . . 7 ⊢ (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵))
25	22, 24	sstrdi 3925	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26	25	sspwd 4573	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
27		ssralv 3982	. . . . 5 ⊢ (𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)) → (∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) → ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))
28	26, 27	syl 18	. . . 4 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) → ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))
29	20, 28	sylbird 263	. . 3 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦) → ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))
30	3, 29	syl5 35	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))
31		simpll 779	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 ∈ UFL)
32		simplr 781	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 = ∪ 𝐵)
33		eqidd 2764	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
34		velpw 4565	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
35		unieq 4881	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
36	35	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑋 = ∪ 𝑥 ↔ 𝑋 = ∪ 𝑧))
37		pweq 4574	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
38	37	ineq1d 4119	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
39	38	rexeqdv 3324	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦))
40	36, 39	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) ↔ (𝑋 = ∪ 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)))
41	40	rspccv 3577	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) → (𝑧 ∈ 𝒫 𝐵 → (𝑋 = ∪ 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)))
42	41	adantl 487	. . . . . . 7 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (𝑧 ∈ 𝒫 𝐵 → (𝑋 = ∪ 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)))
43	34, 42	biimtrrid 246	. . . . . 6 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (𝑧 ⊆ 𝐵 → (𝑋 = ∪ 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)))
44	43	imp32 424	. . . . 5 ⊢ ((((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) ∧ (𝑧 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑧)) → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)
45		unieq 4881	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
46	45	eqeq2d 2774	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑤))
47	46	cbvrexvw 3245	. . . . 5 ⊢ (∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑤)
48	44, 47	sylib 221	. . . 4 ⊢ ((((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) ∧ (𝑧 ⊆ 𝐵 ∧ 𝑋 = ∪ 𝑧)) → ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑤)
49	31, 32, 33, 48	alexsub 24394	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
50	49	ex 418	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) → (topGen‘(fi‘𝐵)) ∈ Comp))
51	30, 50	impbid 215	1 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ∪ cuni 4870  ‘cfv 6558  Fincfn 9001  ficfi 9429  topGenctg 17637  Topctop 23241  TopBasesctb 23293  Compccmp 23734  UFLcufl 24249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-1o 8504  df-2o 8505  df-en 9002  df-dom 9003  df-fin 9005  df-fi 9430  df-topgen 17643  df-fbas 21705  df-fg 21706  df-top 23242  df-topon 23259  df-bases 23294  df-cld 23367  df-ntr 23368  df-cls 23369  df-nei 23446  df-cmp 23735  df-fil 24195  df-ufil 24250  df-ufl 24251  df-flim 24288  df-fcls 24290

This theorem is used by: (None)