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Theorem alexsubb 22049
Description: Biconditional form of the Alexander Subbase Theorem alexsub 22048. (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 2758 . . . . 5 (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵))
21iscmp 21391 . . . 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 483 . . 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 3350 . . . . . . . . . . . 12 (𝑋 ∈ UFL → 𝑋 ∈ V)
65adantr 472 . . . . . . . . . . 11 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 ∈ V)
74, 6eqeltrrd 2838 . . . . . . . . . 10 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
8 uniexb 7136 . . . . . . . . . 10 (𝐵 ∈ V ↔ 𝐵 ∈ V)
97, 8sylibr 224 . . . . . . . . 9 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ∈ V)
10 fiuni 8497 . . . . . . . . 9 (𝐵 ∈ V → 𝐵 = (fi‘𝐵))
119, 10syl 17 . . . . . . . 8 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 = (fi‘𝐵))
12 fibas 20981 . . . . . . . . 9 (fi‘𝐵) ∈ TopBases
13 unitg 20971 . . . . . . . . 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 2802 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝑋 = (topGen‘(fi‘𝐵)))
1615eqeq1d 2760 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑥 (topGen‘(fi‘𝐵)) = 𝑥))
1715eqeq1d 2760 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (𝑋 = 𝑦 (topGen‘(fi‘𝐵)) = 𝑦))
1817rexbidv 3188 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) (topGen‘(fi‘𝐵)) = 𝑦))
1916, 18imbi12d 333 . . . . 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 3122 . . . 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 8488 . . . . . . . 8 (𝐵 ∈ V → 𝐵 ⊆ (fi‘𝐵))
229, 21syl 17 . . . . . . 7 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (fi‘𝐵))
23 bastg 20970 . . . . . . . 8 ((fi‘𝐵) ∈ TopBases → (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵)))
2412, 23ax-mp 5 . . . . . . 7 (fi‘𝐵) ⊆ (topGen‘(fi‘𝐵))
2522, 24syl6ss 3754 . . . . . 6 ((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) → 𝐵 ⊆ (topGen‘(fi‘𝐵)))
26 sspwb 5064 . . . . . 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 3805 . . . . 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 807 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → 𝑋 ∈ UFL)
33 simplr 809 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → 𝑋 = 𝐵)
34 eqidd 2759 . . . 4 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) = (topGen‘(fi‘𝐵)))
35 selpw 4307 . . . . . . 7 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
36 unieq 4594 . . . . . . . . . . 11 (𝑥 = 𝑧 𝑥 = 𝑧)
3736eqeq2d 2768 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑋 = 𝑥𝑋 = 𝑧))
38 pweq 4303 . . . . . . . . . . . 12 (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
3938ineq1d 3954 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑧 ∩ Fin))
4039rexeqdv 3282 . . . . . . . . . 10 (𝑥 = 𝑧 → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦))
4137, 40imbi12d 333 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦) ↔ (𝑋 = 𝑧 → ∃𝑦 ∈ (𝒫 𝑧 ∩ Fin)𝑋 = 𝑦)))
4241rspccv 3444 . . . . . . . 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 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 448 . . . . 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 4594 . . . . . . 7 (𝑦 = 𝑤 𝑦 = 𝑤)
4746eqeq2d 2768 . . . . . 6 (𝑦 = 𝑤 → (𝑋 = 𝑦𝑋 = 𝑤))
4847cbvrexv 3309 . . . . 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 22048 . . 3 (((𝑋 ∈ UFL ∧ 𝑋 = 𝐵) ∧ ∀𝑥 ∈ 𝒫 𝐵(𝑋 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)) → (topGen‘(fi‘𝐵)) ∈ Comp)
5150ex 449 . 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 383   = wceq 1630  wcel 2137  wral 3048  wrex 3049  Vcvv 3338  cin 3712  wss 3713  𝒫 cpw 4300   cuni 4586  cfv 6047  Fincfn 8119  ficfi 8479  topGenctg 16298  Topctop 20898  TopBasesctb 20949  Compccmp 21389  UFLcufl 21903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-fi 8480  df-topgen 16304  df-fbas 19943  df-fg 19944  df-top 20899  df-topon 20916  df-bases 20950  df-cld 21023  df-ntr 21024  df-cls 21025  df-nei 21102  df-cmp 21390  df-fil 21849  df-ufil 21904  df-ufl 21905  df-flim 21942  df-fcls 21944
This theorem is referenced by: (None)
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