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Theorem cmppcmp 31008
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 21919 . 2 (𝐽 ∈ Comp → 𝐽 ∈ Top)
2 cmpcref 31000 . . . . . 6 Comp = CovHasRefFin
32eleq2i 2908 . . . . 5 (𝐽 ∈ Comp ↔ 𝐽 ∈ CovHasRefFin)
4 eqid 2825 . . . . . 6 𝐽 = 𝐽
54iscref 30994 . . . . 5 (𝐽 ∈ CovHasRefFin ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
63, 5bitri 276 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦)))
76simprbi 497 . . 3 (𝐽 ∈ Comp → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦))
8 simprl 767 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ Fin))
9 elin 4172 . . . . . . . . . . . 12 (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑧 ∈ 𝒫 𝐽𝑧 ∈ Fin))
108, 9sylib 219 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ 𝒫 𝐽𝑧 ∈ Fin))
1110simpld 495 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ 𝒫 𝐽)
121ad3antrrr 726 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 ∈ Top)
1310simprd 496 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ Fin)
14 simplr 765 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 = 𝑦)
15 simprr 769 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧Ref𝑦)
16 eqid 2825 . . . . . . . . . . . . . 14 𝑧 = 𝑧
17 eqid 2825 . . . . . . . . . . . . . 14 𝑦 = 𝑦
1816, 17refbas 22034 . . . . . . . . . . . . 13 (𝑧Ref𝑦 𝑦 = 𝑧)
1915, 18syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑦 = 𝑧)
2014, 19eqtrd 2860 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝐽 = 𝑧)
214, 16finlocfin 22044 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑧 ∈ Fin ∧ 𝐽 = 𝑧) → 𝑧 ∈ (LocFin‘𝐽))
2212, 13, 20, 21syl3anc 1365 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (LocFin‘𝐽))
2311, 22elind 4174 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → 𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)))
2423, 15jca 512 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) ∧ (𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦)) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦))
2524ex 413 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) → ((𝑧 ∈ (𝒫 𝐽 ∩ Fin) ∧ 𝑧Ref𝑦) → (𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽)) ∧ 𝑧Ref𝑦)))
2625reximdv2 3275 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑦 ∈ 𝒫 𝐽) ∧ 𝐽 = 𝑦) → (∃𝑧 ∈ (𝒫 𝐽 ∩ Fin)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦))
2726ex 413 . . . . 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 3181 . . 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 31007 . . 3 (𝐽 ∈ Paracomp ↔ 𝐽 ∈ CovHasRef(LocFin‘𝐽))
324iscref 30994 . . 3 (𝐽 ∈ CovHasRef(LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
3331, 32bitri 276 . 2 (𝐽 ∈ Paracomp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝐽 ∩ (LocFin‘𝐽))𝑧Ref𝑦)))
341, 30, 33sylanbrc 583 1 (𝐽 ∈ Comp → 𝐽 ∈ Paracomp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3142  wrex 3143  cin 3938  𝒫 cpw 4541   cuni 4836   class class class wbr 5062  cfv 6351  Fincfn 8501  Topctop 21417  Compccmp 21910  Refcref 22026  LocFinclocfin 22028  CovHasRefccref 30992  Paracompcpcmp 31005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-reg 9048  ax-inf2 9096  ax-ac2 9877
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-fin 8505  df-r1 9185  df-rank 9186  df-card 9360  df-ac 9534  df-top 21418  df-cmp 21911  df-ref 22029  df-locfin 22031  df-cref 30993  df-pcmp 31006
This theorem is referenced by: (None)
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