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Theorem cmppcmp 30259
Description: Every compact space is paracompact. (Contributed by Thierry Arnoux, 7-Jan-2020.)
Assertion
Ref Expression
cmppcmp (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)

Proof of Theorem cmppcmp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmptop 21418 . 2 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 cmpcref 30251 . . . . . 6 Comp = CovHasRefFin
32eleq2i 2841 . . . . 5 (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4 eqid 2770 . . . . . 6 𝐽 = 𝐽
54iscref 30245 . . . . 5 (𝐽 ∈ CovHasRefFin ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
63, 5bitri 264 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
76simprbi 478 . . 3 (𝐽 ∈ Comp → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦))
8 simprl 746 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ Fin))
9 elin 3945 . . . . . . . . . . . 12 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐽𝑧 ∈ Fin))
108, 9sylib 208 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ 𝒫 𝐽𝑧 ∈ Fin))
1110simpld 476 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ 𝒫 𝐽)
121ad3antrrr 701 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 ∈ Top)
1310simprd 477 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ Fin)
14 simplr 744 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 = 𝑦)
15 simprr 748 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧Ref𝑦)
16 eqid 2770 . . . . . . . . . . . . . 14 𝑧 = 𝑧
17 eqid 2770 . . . . . . . . . . . . . 14 𝑦 = 𝑦
1816, 17refbas 21533 . . . . . . . . . . . . 13 (𝑧Ref𝑦 𝑦 = 𝑧)
1915, 18syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑦 = 𝑧)
2014, 19eqtrd 2804 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 = 𝑧)
214, 16finlocfin 21543 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ 𝐽 = 𝑧) → 𝑧 ∈ (LocFin‘𝐽))
2212, 13, 20, 21syl3anc 1475 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (LocFin‘𝐽))
2311, 22elind 3947 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)))
2423, 15jca 495 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦))
2524ex 397 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) → ((𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦)))
2625reximdv2 3161 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦))
2726ex 397 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑦 → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
2827a2d 29 . . . 4 ((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
2928ralimdva 3110 . . 3 (𝐽 ∈ Comp → (∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦) → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
307, 29mpd 15 . 2 (𝐽 ∈ Comp → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦))
31 ispcmp 30258 . . 3 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
324iscref 30245 . . 3 (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
3331, 32bitri 264 . 2 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
341, 30, 33sylanbrc 564 1 (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  cin 3720  𝒫 cpw 4295   cuni 4572   class class class wbr 4784  cfv 6031  Fincfn 8108  Topctop 20917  Compccmp 21409  Refcref 21525  LocFinclocfin 21527  CovHasRefccref 30243  Paracompcpcmp 30256
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 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-reg 8652  ax-inf2 8701  ax-ac2 9486
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  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-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-fin 8112  df-r1 8790  df-rank 8791  df-card 8964  df-ac 9138  df-top 20918  df-cmp 21410  df-ref 21528  df-locfin 21530  df-cref 30244  df-pcmp 30257
This theorem is referenced by: (None)
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