Theorem alexsubb 23962

Description: Biconditional form of the Alexander Subbase Theorem alexsub 23961. (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 2733	. . . . 5 ⊢ ∪ (topGen‘(fi‘𝐵)) = ∪ (topGen‘(fi‘𝐵))
2	1	iscmp 23304	. . . 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 496	. . 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 3458	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ UFL → 𝑋 ∈ V)
6	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 ∈ V)
7	4, 6	eqeltrrd 2834	. . . . . . . . . 10 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 ∈ V)
8		uniexb 7703	. . . . . . . . . 10 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
9	7, 8	sylibr 234	. . . . . . . . 9 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ∈ V)
10		fiuni 9319	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 = ∪ (fi‘𝐵))
12		fibas 22893	. . . . . . . . 9 ⊢ (fi‘𝐵) ∈ TopBases
13		unitg 22883	. . . . . . . . 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 2778	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ (topGen‘(fi‘𝐵)))
16	15	eqeq1d 2735	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑥 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥))
17	15	eqeq1d 2735	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑦 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
18	17	rexbidv 3157	. . . . . 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 3156	. . . 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 9310	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
22	9, 21	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23		bastg 22882	. . . . . . . 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 3943	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26	25	sspwd 4562	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
27		ssralv 3999	. . . . 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 766	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 ∈ UFL)
32		simplr 768	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 = ∪ 𝐵)
33		eqidd 2734	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
34		velpw 4554	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
35		unieq 4869	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
36	35	eqeq2d 2744	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑋 = ∪ 𝑥 ↔ 𝑋 = ∪ 𝑧))
37		pweq 4563	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
38	37	ineq1d 4168	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
39	38	rexeqdv 3294	. . . . . . . . . 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 3570	. . . . . . . 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 4869	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
46	45	eqeq2d 2744	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑤))
47	46	cbvrexvw 3212	. . . . 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 23961	. . 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 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  𝒫 cpw 4549  ∪ cuni 4858  ‘cfv 6486  Fincfn 8875  ficfi 9301  topGenctg 17343  Topctop 22809  TopBasesctb 22861  Compccmp 23302  UFLcufl 23816

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 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-1o 8391  df-2o 8392  df-en 8876  df-dom 8877  df-fin 8879  df-fi 9302  df-topgen 17349  df-fbas 21290  df-fg 21291  df-top 22810  df-topon 22827  df-bases 22862  df-cld 22935  df-ntr 22936  df-cls 22937  df-nei 23014  df-cmp 23303  df-fil 23762  df-ufil 23817  df-ufl 23818  df-flim 23855  df-fcls 23857

This theorem is referenced by: (None)