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Theorem pcmplfinf 32433
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 1213 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑈𝐽)
3 simpll3 1214 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑋 = 𝑈)
4 elpwi 4567 . . . 4 (𝑣 ∈ 𝒫 𝐽𝑣𝐽)
54ad2antlr 725 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣𝐽)
6 simprr 771 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣Ref𝑈)
7 simprl 769 . . 3 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣 ∈ (LocFin‘𝐽))
81, 2, 3, 5, 6, 7locfinref 32413 . 2 ((((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
91pcmplfin 32432 . 2 ((𝐽 ∈ Paracomp ∧ 𝑈𝐽𝑋 = 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
108, 9r19.29a 3159 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wss 3910  𝒫 cpw 4560   cuni 4865   class class class wbr 5105  ran crn 5634  wf 6492  cfv 6496  Refcref 22851  LocFinclocfin 22853  Paracompcpcmp 32427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-reg 9527  ax-inf2 9576  ax-ac2 10398
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7312  df-ov 7359  df-om 7802  df-2nd 7921  df-frecs 8211  df-wrecs 8242  df-recs 8316  df-rdg 8355  df-1o 8411  df-er 8647  df-en 8883  df-dom 8884  df-fin 8886  df-r1 9699  df-rank 9700  df-card 9874  df-ac 10051  df-top 22241  df-topon 22258  df-ref 22854  df-locfin 22856  df-cref 32415  df-pcmp 32428
This theorem is referenced by: (None)
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