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Theorem alexsubb 24070
Description: Biconditional form of the Alexander Subbase Theorem alexsub 24069. (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.
StepHypRef Expression
1 eqid 2735 . . . . 5 (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵))
21iscmp 23412 . . . 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)))
32simprbi 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 3499 . . . . . . . . . . . 12 (𝑋 ∈ UFL → 𝑋 ∈ V)
65adantr 480 . . . . . . . . . . 11 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 ∈ V)
74, 6eqeltrrd 2840 . . . . . . . . . 10 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
8 uniexb 7783 . . . . . . . . . 10 (𝐵 ∈ V ↔ 𝐵 ∈ V)
97, 8sylibr 234 . . . . . . . . 9 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
10 fiuni 9466 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
119, 10syl 17 . . . . . . . 8 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 = (fi‘𝐵))
12 fibas 23000 . . . . . . . . 9 (fi‘𝐵) ∈ TopBases
13 unitg 22990 . . . . . . . . 9 ((fi‘𝐵) ∈ TopBases → (topGen‘(fi‘𝐵)) = (fi‘𝐵))
1412, 13mp1i 13 . . . . . . . 8 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (topGen‘(fi‘𝐵)) = (fi‘𝐵))
1511, 4, 143eqtr4d 2785 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 = (topGen‘(fi‘𝐵)))
1615eqeq1d 2737 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑥 (topGen‘(fi‘𝐵)) = 𝑥))
1715eqeq1d 2737 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑦 (topGen‘(fi‘𝐵)) = 𝑦))
1817rexbidv 3177 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) (topGen‘(fi‘𝐵)) = 𝑦))
1916, 18imbi12d 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)))
2019ralbidv 3176 . . . 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 9457 . . . . . . . 8 (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
229, 21syl 17 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23 bastg 22989 . . . . . . . 8 ((fi‘𝐵) ∈ TopBases → (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵)))
2412, 23ax-mp 5 . . . . . . 7 (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵))
2522, 24sstrdi 4008 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
2625sspwd 4618 . . . . 5 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
27 ssralv 4064 . . . . 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)))
2826, 27syl 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)))
2920, 28sylbird 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)))
303, 29syl5 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 2736 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
34 velpw 4610 . . . . . . 7 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
35 unieq 4923 . . . . . . . . . . 11 (𝑥 = 𝑧 𝑥 = 𝑧)
3635eqeq2d 2746 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑋 = 𝑥𝑋 = 𝑧))
37 pweq 4619 . . . . . . . . . . . 12 (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
3837ineq1d 4227 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
3938rexeqdv 3325 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦))
4036, 39imbi12d 344 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) ↔ (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4140rspccv 3619 . . . . . . . 8 (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (𝑧 ∈ 𝒫 𝐵 → (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4241adantl 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)))
4334, 42biimtrrid 243 . . . . . 6 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (𝑧𝐵 → (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4443imp32 418 . . . . 5 ((((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) ∧ (𝑧𝐵𝑋 = 𝑧)) → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)
45 unieq 4923 . . . . . . 7 (𝑦 = 𝑤 𝑦 = 𝑤)
4645eqeq2d 2746 . . . . . 6 (𝑦 = 𝑤 → (𝑋 = 𝑦𝑋 = 𝑤))
4746cbvrexvw 3236 . . . . 5 (∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑤)
4844, 47sylib 218 . . . 4 ((((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) ∧ (𝑧𝐵𝑋 = 𝑧)) → ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑤)
4931, 32, 33, 48alexsub 24069 . . 3 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
5049ex 412 . 2 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (topGen‘(fi‘𝐵)) ∈ Comp))
5130, 50impbid 212 1 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  cfv 6563  Fincfn 8984  ficfi 9448  topGenctg 17484  Topctop 22915  TopBasesctb 22968  Compccmp 23410  UFLcufl 23924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-topgen 17490  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cmp 23411  df-fil 23870  df-ufil 23925  df-ufl 23926  df-flim 23963  df-fcls 23965
This theorem is referenced by: (None)
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