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Theorem pcmplfinf 32841
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 1214 . . 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 1215 . . 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 726 . . 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 772 . . 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 770 . . 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 32821 . 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 32840 . 2 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ))
108, 9r19.29a 3163 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  Refcref 23006  LocFinclocfin 23008  Paracompcpcmp 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636  ax-ac2 10458
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-fin 8943  df-r1 9759  df-rank 9760  df-card 9934  df-ac 10111  df-top 22396  df-topon 22413  df-ref 23009  df-locfin 23011  df-cref 32823  df-pcmp 32836
This theorem is referenced by: (None)
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