Theorem alexsubb 21850

Description: Biconditional form of the Alexander Subbase Theorem alexsub 21849. (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 2622	. . . . 5 ⊢ ∪ (topGen‘(fi‘𝐵)) = ∪ (topGen‘(fi‘𝐵))
2	1	iscmp 21191	. . . 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 480	. . 3 ⊢ ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))(∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
4		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ 𝐵)
5		elex 3212	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ UFL → 𝑋 ∈ V)
6	5	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 ∈ V)
7	4, 6	eqeltrrd 2702	. . . . . . . . . 10 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 ∈ V)
8		uniexb 6973	. . . . . . . . . 10 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
9	7, 8	sylibr 224	. . . . . . . . 9 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ∈ V)
10		fiuni 8334	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 = ∪ (fi‘𝐵))
12		fibas 20781	. . . . . . . . 9 ⊢ (fi‘𝐵) ∈ TopBases
13		unitg 20771	. . . . . . . . 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 2666	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ (topGen‘(fi‘𝐵)))
16	15	eqeq1d 2624	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑥 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥))
17	15	eqeq1d 2624	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑦 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
18	17	rexbidv 3052	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
19	16, 18	imbi12d 334	. . . . 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 2986	. . . 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 8325	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
22	9, 21	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23		bastg 20770	. . . . . . . 8 ⊢ ((fi‘𝐵) ∈ TopBases → (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵)))
24	12, 23	ax-mp 5	. . . . . . 7 ⊢ (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵))
25	22, 24	syl6ss 3615	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26		sspwb 4917	. . . . . 6 ⊢ (𝐵 ⊆ (topGen‘(fi‘𝐵)) ↔ 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
27	25, 26	sylib 208	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
28		ssralv 3666	. . . . 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)))
29	27, 28	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)))
30	20, 29	sylbird 250	. . 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)))
31	3, 30	syl5 34	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)))
32		simpll 790	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 ∈ UFL)
33		simplr 792	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → 𝑋 = ∪ 𝐵)
34		eqidd 2623	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
35		selpw 4165	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
36		unieq 4444	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
37	36	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑋 = ∪ 𝑥 ↔ 𝑋 = ∪ 𝑧))
38		pweq 4161	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
39	38	ineq1d 3813	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
40	39	rexeqdv 3145	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦))
41	37, 40	imbi12d 334	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) ↔ (𝑋 = ∪ 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)))
42	41	rspccv 3306	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) → (𝑧 ∈ 𝒫 𝐵 → (𝑋 = ∪ 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦)))
43	42	adantl 482	. . . . . . 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)))
44	35, 43	syl5bir 233	. . . . . 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)))
45	44	imp32 449	. . . . 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)
46		unieq 4444	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
47	46	eqeq2d 2632	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑤))
48	47	cbvrexv 3172	. . . . 5 ⊢ (∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑦 ↔ ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = ∪ 𝑤)
49	45, 48	sylib 208	. . . 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)
50	32, 33, 34, 49	alexsub 21849	. . 3 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
51	50	ex 450	. 2 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦) → (topGen‘(fi‘𝐵)) ∈ Comp))
52	31, 51	impbid 202	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436  ‘cfv 5888  Fincfn 7955  ficfi 8316  topGenctg 16098  Topctop 20698  TopBasesctb 20749  Compccmp 21189  UFLcufl 21704

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-rep 4771  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-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-topgen 16104  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cmp 21190  df-fil 21650  df-ufil 21705  df-ufl 21706  df-flim 21743  df-fcls 21745

This theorem is referenced by: (None)