Theorem alexsubb 24011

Description: Biconditional form of the Alexander Subbase Theorem alexsub 24010. (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 2736	. . . . 5 ⊢ ∪ (topGen‘(fi‘𝐵)) = ∪ (topGen‘(fi‘𝐵))
2	1	iscmp 23353	. . . 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 497	. . 3 ⊢ ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
4		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ 𝐵)
5		elex 3450	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ UFL → 𝑋 ∈ V)
6	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 ∈ V)
7	4, 6	eqeltrrd 2837	. . . . . . . . . 10 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 ∈ V)
8		uniexb 7718	. . . . . . . . . 10 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
9	7, 8	sylibr 234	. . . . . . . . 9 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ∈ V)
10		fiuni 9341	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 = ∪ (fi‘𝐵))
12		fibas 22942	. . . . . . . . 9 ⊢ (fi‘𝐵) ∈ TopBases
13		unitg 22932	. . . . . . . . 9 ⊢ ((fi‘𝐵) ∈ TopBases → ∪ (topGen‘(fi‘𝐵)) = ∪ (fi‘𝐵))
14	12, 13	mp1i 13	. . . . . . . 8 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ (topGen‘(fi‘𝐵)) = ∪ (fi‘𝐵))
15	11, 4, 14	3eqtr4d 2781	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ (topGen‘(fi‘𝐵)))
16	15	eqeq1d 2738	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑥 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥))
17	15	eqeq1d 2738	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑦 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
18	17	rexbidv 3161	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
19	16, 18	imbi12d 344	. . . . 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 3160	. . . 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 9332	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
22	9, 21	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23		bastg 22931	. . . . . . . 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 3934	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26	25	sspwd 4554	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
27		ssralv 3990	. . . . 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 17	. . . 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 260	. . 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 34	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))
31		simpll 767	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 ∈ UFL)
32		simplr 769	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 = ∪ 𝐵)
33		eqidd 2737	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
34		velpw 4546	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
35		unieq 4861	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
36	35	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑋 = ∪ 𝑥 ↔ 𝑋 = ∪ 𝑧))
37		pweq 4555	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
38	37	ineq1d 4159	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
39	38	rexeqdv 3296	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦))
40	36, 39	imbi12d 344	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) ↔ (𝑋 = ∪ 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)))
41	40	rspccv 3561	. . . . . . . 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 481	. . . . . . 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 243	. . . . . 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 418	. . . . 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 4861	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
46	45	eqeq2d 2747	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑤))
47	46	cbvrexvw 3216	. . . . 5 ⊢ (∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑤)
48	44, 47	sylib 218	. . . 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 24010	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
50	49	ex 412	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) → (topGen‘(fi‘𝐵)) ∈ Comp))
51	30, 50	impbid 212	1 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  𝒫 cpw 4541  ∪ cuni 4850  ‘cfv 6498  Fincfn 8893  ficfi 9323  topGenctg 17400  Topctop 22858  TopBasesctb 22910  Compccmp 23351  UFLcufl 23865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-1o 8405  df-2o 8406  df-en 8894  df-dom 8895  df-fin 8897  df-fi 9324  df-topgen 17406  df-fbas 21349  df-fg 21350  df-top 22859  df-topon 22876  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986  df-nei 23063  df-cmp 23352  df-fil 23811  df-ufil 23866  df-ufl 23867  df-flim 23904  df-fcls 23906

This theorem is referenced by: (None)