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Theorem alexsubALTlem1 22048
Description: Lemma for alexsubALT 22052. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)
Hypothesis
Ref Expression
alexsubALT.1 𝑋 = 𝐽
Assertion
Ref Expression
alexsubALTlem1 (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
Distinct variable groups:   𝑐,𝑑,𝑥,𝐽   𝑋,𝑐,𝑑,𝑥

Proof of Theorem alexsubALTlem1
StepHypRef Expression
1 cmptop 21396 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 fitop 20903 . . . . 5 (𝐽 ∈ Top → (fi‘𝐽) = 𝐽)
32fveq2d 6352 . . . 4 (𝐽 ∈ Top → (topGen‘(fi‘𝐽)) = (topGen‘𝐽))
4 tgtop 20975 . . . 4 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
53, 4eqtr2d 2791 . . 3 (𝐽 ∈ Top → 𝐽 = (topGen‘(fi‘𝐽)))
61, 5syl 17 . 2 (𝐽 ∈ Comp → 𝐽 = (topGen‘(fi‘𝐽)))
7 selpw 4305 . . . 4 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
8 alexsubALT.1 . . . . . 6 𝑋 = 𝐽
98cmpcov 21390 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
1093exp 1113 . . . 4 (𝐽 ∈ Comp → (𝑐𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
117, 10syl5bi 232 . . 3 (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
1211ralrimiv 3099 . 2 (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
13 fveq2 6348 . . . . . 6 (𝑥 = 𝐽 → (fi‘𝑥) = (fi‘𝐽))
1413fveq2d 6352 . . . . 5 (𝑥 = 𝐽 → (topGen‘(fi‘𝑥)) = (topGen‘(fi‘𝐽)))
1514eqeq2d 2766 . . . 4 (𝑥 = 𝐽 → (𝐽 = (topGen‘(fi‘𝑥)) ↔ 𝐽 = (topGen‘(fi‘𝐽))))
16 pweq 4301 . . . . 5 (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
1716raleqdv 3279 . . . 4 (𝑥 = 𝐽 → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑) ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
1815, 17anbi12d 749 . . 3 (𝑥 = 𝐽 → ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)) ↔ (𝐽 = (topGen‘(fi‘𝐽)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
1918spcegv 3430 . 2 (𝐽 ∈ Comp → ((𝐽 = (topGen‘(fi‘𝐽)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)) → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))))
206, 12, 19mp2and 717 1 (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wex 1849  wcel 2135  wral 3046  wrex 3047  cin 3710  wss 3711  𝒫 cpw 4298   cuni 4584  cfv 6045  Fincfn 8117  ficfi 8477  topGenctg 16296  Topctop 20896  Compccmp 21387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-en 8118  df-fin 8121  df-fi 8478  df-topgen 16302  df-top 20897  df-cmp 21388
This theorem is referenced by:  alexsubALT  22052
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