Theorem alexsubb 24021

Description: Biconditional form of the Alexander Subbase Theorem alexsub 24020. (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 2737	. . . . 5 ⊢ ∪ (topGen‘(fi‘𝐵)) = ∪ (topGen‘(fi‘𝐵))
2	1	iscmp 23363	. . . 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 3451	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ UFL → 𝑋 ∈ V)
6	5	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 ∈ V)
7	4, 6	eqeltrrd 2838	. . . . . . . . . 10 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 ∈ V)
8		uniexb 7711	. . . . . . . . . 10 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V)
9	7, 8	sylibr 234	. . . . . . . . 9 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ∈ V)
10		fiuni 9334	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ∪ 𝐵 = ∪ (fi‘𝐵))
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → ∪ 𝐵 = ∪ (fi‘𝐵))
12		fibas 22952	. . . . . . . . 9 ⊢ (fi‘𝐵) ∈ TopBases
13		unitg 22942	. . . . . . . . 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 2782	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝑋 = ∪ (topGen‘(fi‘𝐵)))
16	15	eqeq1d 2739	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑥 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑥))
17	15	eqeq1d 2739	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → (𝑋 = ∪ 𝑦 ↔ ∪ (topGen‘(fi‘𝐵)) = ∪ 𝑦))
18	17	rexbidv 3162	. . . . . 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 3161	. . . 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 9325	. . . . . . . 8 ⊢ (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
22	9, 21	syl 17	. . . . . . 7 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23		bastg 22941	. . . . . . . 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 3935	. . . . . 6 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26	25	sspwd 4555	. . . . 5 ⊢ ((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
27		ssralv 3991	. . . . 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 2738	. . . 4 ⊢ (((𝑋 ∈ UFL ∧ 𝑋 = ∪ 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = ∪ 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = ∪ 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
34		velpw 4547	. . . . . . 7 ⊢ (𝑧 ∈ 𝒫 𝐵 ↔ 𝑧 ⊆ 𝐵)
35		unieq 4862	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
36	35	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑋 = ∪ 𝑥 ↔ 𝑋 = ∪ 𝑧))
37		pweq 4556	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
38	37	ineq1d 4160	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
39	38	rexeqdv 3297	. . . . . . . . . 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 3562	. . . . . . . 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 4862	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ∪ 𝑦 = ∪ 𝑤)
46	45	eqeq2d 2748	. . . . . 6 ⊢ (𝑦 = 𝑤 → (𝑋 = ∪ 𝑦 ↔ 𝑋 = ∪ 𝑤))
47	46	cbvrexvw 3217	. . . . 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 24020	. . 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 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851  ‘cfv 6492  Fincfn 8886  ficfi 9316  topGenctg 17391  Topctop 22868  TopBasesctb 22920  Compccmp 23361  UFLcufl 23875

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-1o 8398  df-2o 8399  df-en 8887  df-dom 8888  df-fin 8890  df-fi 9317  df-topgen 17397  df-fbas 21341  df-fg 21342  df-top 22869  df-topon 22886  df-bases 22921  df-cld 22994  df-ntr 22995  df-cls 22996  df-nei 23073  df-cmp 23362  df-fil 23821  df-ufil 23876  df-ufl 23877  df-flim 23914  df-fcls 23916

This theorem is referenced by: (None)