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Theorem alexsubALTlem1 22071
Description: Lemma for alexsubALT 22075. 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 21419 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 fitop 20925 . . . . 5 (𝐽 ∈ Top → (fi‘𝐽) = 𝐽)
32fveq2d 6336 . . . 4 (𝐽 ∈ Top → (topGen‘(fi‘𝐽)) = (topGen‘𝐽))
4 tgtop 20998 . . . 4 (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
53, 4eqtr2d 2806 . . 3 (𝐽 ∈ Top → 𝐽 = (topGen‘(fi‘𝐽)))
61, 5syl 17 . 2 (𝐽 ∈ Comp → 𝐽 = (topGen‘(fi‘𝐽)))
7 selpw 4304 . . . 4 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
8 alexsubALT.1 . . . . . 6 𝑋 = 𝐽
98cmpcov 21413 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
1093exp 1112 . . . 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 3114 . 2 (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
13 fveq2 6332 . . . . . 6 (𝑥 = 𝐽 → (fi‘𝑥) = (fi‘𝐽))
1413fveq2d 6336 . . . . 5 (𝑥 = 𝐽 → (topGen‘(fi‘𝑥)) = (topGen‘(fi‘𝐽)))
1514eqeq2d 2781 . . . 4 (𝑥 = 𝐽 → (𝐽 = (topGen‘(fi‘𝑥)) ↔ 𝐽 = (topGen‘(fi‘𝐽))))
16 pweq 4300 . . . . 5 (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
1716raleqdv 3293 . . . 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 616 . . 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 3445 . 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 679 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 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  cin 3722  wss 3723  𝒫 cpw 4297   cuni 4574  cfv 6031  Fincfn 8109  ficfi 8472  topGenctg 16306  Topctop 20918  Compccmp 21410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-fin 8113  df-fi 8473  df-topgen 16312  df-top 20919  df-cmp 21411
This theorem is referenced by:  alexsubALT  22075
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