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Theorem alexsubb 21769
Description: Biconditional form of the Alexander Subbase Theorem alexsub 21768. (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 2621 . . . . 5 (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵))
21iscmp 21110 . . . 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 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 3201 . . . . . . . . . . . 12 (𝑋 ∈ UFL → 𝑋 ∈ V)
65adantr 481 . . . . . . . . . . 11 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 ∈ V)
74, 6eqeltrrd 2699 . . . . . . . . . 10 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
8 uniexb 6928 . . . . . . . . . 10 (𝐵 ∈ V ↔ 𝐵 ∈ V)
97, 8sylibr 224 . . . . . . . . 9 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
10 fiuni 8285 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
119, 10syl 17 . . . . . . . 8 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 = (fi‘𝐵))
12 fibas 20701 . . . . . . . . 9 (fi‘𝐵) ∈ TopBases
13 unitg 20691 . . . . . . . . 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 2665 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 = (topGen‘(fi‘𝐵)))
1615eqeq1d 2623 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑥 (topGen‘(fi‘𝐵)) = 𝑥))
1715eqeq1d 2623 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑦 (topGen‘(fi‘𝐵)) = 𝑦))
1817rexbidv 3046 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) (topGen‘(fi‘𝐵)) = 𝑦))
1916, 18imbi12d 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)))
2019ralbidv 2981 . . . 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 8276 . . . . . . . 8 (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
229, 21syl 17 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23 bastg 20690 . . . . . . . 8 ((fi‘𝐵) ∈ TopBases → (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵)))
2412, 23ax-mp 5 . . . . . . 7 (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵))
2522, 24syl6ss 3599 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26 sspwb 4883 . . . . . 6 (𝐵 ⊆ (topGen‘(fi‘𝐵)) ↔ 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
2725, 26sylib 208 . . . . 5 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝒫 𝐵 ⊆ 𝒫 (topGen‘(fi‘𝐵)))
28 ssralv 3650 . . . . 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 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)))
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 789 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → 𝑋 ∈ UFL)
33 simplr 791 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → 𝑋 = 𝐵)
34 eqidd 2622 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
35 selpw 4142 . . . . . . 7 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
36 unieq 4415 . . . . . . . . . . 11 (𝑥 = 𝑧 𝑥 = 𝑧)
3736eqeq2d 2631 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑋 = 𝑥𝑋 = 𝑧))
38 pweq 4138 . . . . . . . . . . . 12 (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
3938ineq1d 3796 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
4039rexeqdv 3137 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦))
4137, 40imbi12d 334 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) ↔ (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4241rspccv 3295 . . . . . . . 8 (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (𝑧 ∈ 𝒫 𝐵 → (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4342adantl 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)))
4435, 43syl5bir 233 . . . . . 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 449 . . . . 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 4415 . . . . . . 7 (𝑦 = 𝑤 𝑦 = 𝑤)
4746eqeq2d 2631 . . . . . 6 (𝑦 = 𝑤 → (𝑋 = 𝑦𝑋 = 𝑤))
4847cbvrexv 3163 . . . . 5 (∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑤 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑤)
4945, 48sylib 208 . . . 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 21768 . . 3 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
5150ex 450 . 2 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) → (topGen‘(fi‘𝐵)) ∈ Comp))
5231, 51impbid 202 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 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  cin 3558  wss 3559  𝒫 cpw 4135   cuni 4407  cfv 5852  Fincfn 7906  ficfi 8267  topGenctg 16026  Topctop 20626  TopBasesctb 20669  Compccmp 21108  UFLcufl 21623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-fi 8268  df-topgen 16032  df-fbas 19671  df-fg 19672  df-top 20627  df-topon 20644  df-bases 20670  df-cld 20742  df-ntr 20743  df-cls 20744  df-nei 20821  df-cmp 21109  df-fil 21569  df-ufil 21624  df-ufl 21625  df-flim 21662  df-fcls 21664
This theorem is referenced by: (None)
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