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Theorem alexsubb 22129
Description: Biconditional form of the Alexander Subbase Theorem alexsub 22128. (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 2765 . . . . 5 (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵))
21iscmp 21471 . . . 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 490 . . 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 3365 . . . . . . . . . . . 12 (𝑋 ∈ UFL → 𝑋 ∈ V)
65adantr 472 . . . . . . . . . . 11 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 ∈ V)
74, 6eqeltrrd 2845 . . . . . . . . . 10 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
8 uniexb 7171 . . . . . . . . . 10 (𝐵 ∈ V ↔ 𝐵 ∈ V)
97, 8sylibr 225 . . . . . . . . 9 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
10 fiuni 8541 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
119, 10syl 17 . . . . . . . 8 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 = (fi‘𝐵))
12 fibas 21061 . . . . . . . . 9 (fi‘𝐵) ∈ TopBases
13 unitg 21051 . . . . . . . . 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 2809 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 = (topGen‘(fi‘𝐵)))
1615eqeq1d 2767 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑥 (topGen‘(fi‘𝐵)) = 𝑥))
1715eqeq1d 2767 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑦 (topGen‘(fi‘𝐵)) = 𝑦))
1817rexbidv 3199 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) (topGen‘(fi‘𝐵)) = 𝑦))
1916, 18imbi12d 335 . . . . 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 3133 . . . 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 8532 . . . . . . . 8 (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
229, 21syl 17 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23 bastg 21050 . . . . . . . 8 ((fi‘𝐵) ∈ TopBases → (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵)))
2412, 23ax-mp 5 . . . . . . 7 (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵))
2522, 24syl6ss 3773 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26 sspwb 5073 . . . . . 6 (𝐵 ⊆ (topGen‘(fi‘𝐵)) ↔ 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
2725, 26sylib 209 . . . . 5 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
28 ssralv 3826 . . . . 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)))
2927, 28syl 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)))
3020, 29sylbird 251 . . 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)))
313, 30syl5 34 . 2 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)))
32 simpll 783 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → 𝑋 ∈ UFL)
33 simplr 785 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → 𝑋 = 𝐵)
34 eqidd 2766 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
35 selpw 4322 . . . . . . 7 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
36 unieq 4602 . . . . . . . . . . 11 (𝑥 = 𝑧 𝑥 = 𝑧)
3736eqeq2d 2775 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑋 = 𝑥𝑋 = 𝑧))
38 pweq 4318 . . . . . . . . . . . 12 (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
3938ineq1d 3975 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
4039rexeqdv 3293 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦))
4137, 40imbi12d 335 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) ↔ (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4241rspccv 3458 . . . . . . . 8 (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (𝑧 ∈ 𝒫 𝐵 → (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4342adantl 473 . . . . . . 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)))
4435, 43syl5bir 234 . . . . . 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)))
4544imp32 409 . . . . 5 ((((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) ∧ (𝑧𝐵𝑋 = 𝑧)) → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)
46 unieq 4602 . . . . . . 7 (𝑦 = 𝑤 𝑦 = 𝑤)
4746eqeq2d 2775 . . . . . 6 (𝑦 = 𝑤 → (𝑋 = 𝑦𝑋 = 𝑤))
4847cbvrexv 3320 . . . . 5 (∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑤)
4945, 48sylib 209 . . . 4 ((((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) ∧ (𝑧𝐵𝑋 = 𝑧)) → ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑤)
5032, 33, 34, 49alexsub 22128 . . 3 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
5150ex 401 . 2 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (topGen‘(fi‘𝐵)) ∈ Comp))
5231, 51impbid 203 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cin 3731  wss 3732  𝒫 cpw 4315   cuni 4594  cfv 6068  Fincfn 8160  ficfi 8523  topGenctg 16364  Topctop 20977  TopBasesctb 21029  Compccmp 21469  UFLcufl 21983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fi 8524  df-topgen 16370  df-fbas 20016  df-fg 20017  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-cmp 21470  df-fil 21929  df-ufil 21984  df-ufl 21985  df-flim 22022  df-fcls 22024
This theorem is referenced by: (None)
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