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Theorem alexsubb 22220
 Description: Biconditional form of the Alexander Subbase Theorem alexsub 22219. (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 2825 . . . . 5 (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵))
21iscmp 21562 . . . 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 492 . . 3 ((topGen‘(fi‘𝐵)) ∈ Comp → ∀𝑥 ∈ 𝒫 (topGen‘(fi‘𝐵))( (topGen‘(fi‘𝐵)) = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) (topGen‘(fi‘𝐵)) = 𝑦))
4 simpr 479 . . . . . . . . . . 11 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 = 𝐵)
5 elex 3429 . . . . . . . . . . . 12 (𝑋 ∈ UFL → 𝑋 ∈ V)
65adantr 474 . . . . . . . . . . 11 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 ∈ V)
74, 6eqeltrrd 2907 . . . . . . . . . 10 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
8 uniexb 7233 . . . . . . . . . 10 (𝐵 ∈ V ↔ 𝐵 ∈ V)
97, 8sylibr 226 . . . . . . . . 9 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
10 fiuni 8603 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
119, 10syl 17 . . . . . . . 8 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 = (fi‘𝐵))
12 fibas 21152 . . . . . . . . 9 (fi‘𝐵) ∈ TopBases
13 unitg 21142 . . . . . . . . 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 2871 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 = (topGen‘(fi‘𝐵)))
1615eqeq1d 2827 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑥 (topGen‘(fi‘𝐵)) = 𝑥))
1715eqeq1d 2827 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑦 (topGen‘(fi‘𝐵)) = 𝑦))
1817rexbidv 3262 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) (topGen‘(fi‘𝐵)) = 𝑦))
1916, 18imbi12d 336 . . . . 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 3195 . . . 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 8594 . . . . . . . 8 (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
229, 21syl 17 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23 bastg 21141 . . . . . . . 8 ((fi‘𝐵) ∈ TopBases → (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵)))
2412, 23ax-mp 5 . . . . . . 7 (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵))
2522, 24syl6ss 3839 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26 sspwb 5138 . . . . . 6 (𝐵 ⊆ (topGen‘(fi‘𝐵)) ↔ 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
2725, 26sylib 210 . . . . 5 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
28 ssralv 3891 . . . . 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 252 . . 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 2826 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
35 selpw 4385 . . . . . . 7 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
36 unieq 4666 . . . . . . . . . . 11 (𝑥 = 𝑧 𝑥 = 𝑧)
3736eqeq2d 2835 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑋 = 𝑥𝑋 = 𝑧))
38 pweq 4381 . . . . . . . . . . . 12 (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
3938ineq1d 4040 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
4039rexeqdv 3357 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦))
4137, 40imbi12d 336 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) ↔ (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4241rspccv 3523 . . . . . . . 8 (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (𝑧 ∈ 𝒫 𝐵 → (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4342adantl 475 . . . . . . 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 235 . . . . . 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 411 . . . . 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 4666 . . . . . . 7 (𝑦 = 𝑤 𝑦 = 𝑤)
4746eqeq2d 2835 . . . . . 6 (𝑦 = 𝑤 → (𝑋 = 𝑦𝑋 = 𝑤))
4847cbvrexv 3384 . . . . 5 (∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑤)
4945, 48sylib 210 . . . 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 22219 . . 3 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
5150ex 403 . 2 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (topGen‘(fi‘𝐵)) ∈ Comp))
5231, 51impbid 204 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 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118  Vcvv 3414   ∩ cin 3797   ⊆ wss 3798  𝒫 cpw 4378  ∪ cuni 4658  ‘cfv 6123  Fincfn 8222  ficfi 8585  topGenctg 16451  Topctop 21068  TopBasesctb 21120  Compccmp 21560  UFLcufl 22074 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fi 8586  df-topgen 16457  df-fbas 20103  df-fg 20104  df-top 21069  df-topon 21086  df-bases 21121  df-cld 21194  df-ntr 21195  df-cls 21196  df-nei 21273  df-cmp 21561  df-fil 22020  df-ufil 22075  df-ufl 22076  df-flim 22113  df-fcls 22115 This theorem is referenced by: (None)
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