Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pcmplfinf Structured version   Visualization version   GIF version

Theorem pcmplfinf 29049
Description: Given a paracompact topology 𝐽 and an open cover 𝑈, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover 𝑈. (Contributed by Thierry Arnoux, 31-Jan-2020.)
Hypothesis
Ref Expression
pcmplfin.x 𝑋 = 𝐽
Assertion
Ref Expression
pcmplfinf ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
Distinct variable groups:   𝑓,𝐽   𝑈,𝑓   𝑓,𝑋

Proof of Theorem pcmplfinf
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 pcmplfin.x . . 3 𝑋 = 𝐽
2 simpll2 1093 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑈𝐽)
3 simpll3 1094 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑋 = 𝑈)
4 elpwi 4116 . . . 4 (𝑣 ∈ 𝒫 𝐽𝑣𝐽)
54ad2antlr 758 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣𝐽)
6 simprr 791 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣Ref𝑈)
7 simprl 789 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣 ∈ (LocFin‘𝐽))
81, 2, 3, 5, 6, 7locfinref 29029 . 2 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
91pcmplfin 29048 . 2 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
108, 9r19.29a 3059 1 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wss 3539  𝒫 cpw 4107   cuni 4366   class class class wbr 4577  ran crn 5028  wf 5785  cfv 5789  Refcref 21062  LocFinclocfin 21064  Paracompcpcmp 29043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-reg 8357  ax-inf2 8398  ax-ac2 9145
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-fin 7822  df-r1 8487  df-rank 8488  df-card 8625  df-ac 8799  df-top 20468  df-topon 20470  df-ref 21065  df-locfin 21067  df-cref 29031  df-pcmp 29044
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator