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 33532
Description: Given a paracompact topology 𝐽 and an open cover π‘ˆ, there exists an open refinement ran 𝑓 that is locally finite, using the same index as the original cover π‘ˆ. (Contributed by Thierry Arnoux, 31-Jan-2020.)
Hypothesis
Ref Expression
pcmplfin.x 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
pcmplfinf ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘“(𝑓:π‘ˆβŸΆπ½ ∧ ran 𝑓Refπ‘ˆ ∧ ran 𝑓 ∈ (LocFinβ€˜π½)))
Distinct variable groups:   𝑓,𝐽   π‘ˆ,𝑓   𝑓,𝑋
Proof of Theorem pcmplfinf
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 pcmplfin.x . . 3 𝑋 = βˆͺ 𝐽
2 simpll2 1210 . . 3 ((((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ)) β†’ π‘ˆ βŠ† 𝐽)
3 simpll3 1211 . . 3 ((((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ)) β†’ 𝑋 = βˆͺ π‘ˆ)
4 elpwi 4610 . . . 4 (𝑣 ∈ 𝒫 𝐽 β†’ 𝑣 βŠ† 𝐽)
54ad2antlr 725 . . 3 ((((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ)) β†’ 𝑣 βŠ† 𝐽)
6 simprr 771 . . 3 ((((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ)) β†’ 𝑣Refπ‘ˆ)
7 simprl 769 . . 3 ((((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ)) β†’ 𝑣 ∈ (LocFinβ€˜π½))
81, 2, 3, 5, 6, 7locfinref 33512 . 2 ((((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ)) β†’ βˆƒπ‘“(𝑓:π‘ˆβŸΆπ½ ∧ ran 𝑓Refπ‘ˆ ∧ ran 𝑓 ∈ (LocFinβ€˜π½)))
91pcmplfin 33531 . 2 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ))
108, 9r19.29a 3152 1 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘“(𝑓:π‘ˆβŸΆπ½ ∧ ran 𝑓Refπ‘ˆ ∧ ran 𝑓 ∈ (LocFinβ€˜π½)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098   βŠ† wss 3945  π’« cpw 4603  βˆͺ cuni 4908   class class class wbr 5148  ran crn 5678  βŸΆwf 6543  β€˜cfv 6547  Refcref 23436  LocFinclocfin 23438  Paracompcpcmp 33526
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-reg 9615  ax-inf2 9664  ax-ac2 10486
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7373  df-ov 7420  df-om 7870  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-fin 8966  df-r1 9787  df-rank 9788  df-card 9962  df-ac 10139  df-top 22826  df-topon 22843  df-ref 23439  df-locfin 23441  df-cref 33514  df-pcmp 33527
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator