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Theorem pcmplfinf 33398
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 1211 . . 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 1212 . . 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 4605 . . . 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 33378 . 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 33397 . 2 ((𝐽 ∈ Paracomp ∧ π‘ˆ βŠ† 𝐽 ∧ 𝑋 = βˆͺ π‘ˆ) β†’ βˆƒπ‘£ ∈ 𝒫 𝐽(𝑣 ∈ (LocFinβ€˜π½) ∧ 𝑣Refπ‘ˆ))
108, 9r19.29a 3157 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 395   ∧ w3a 1085   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099   βŠ† wss 3944  π’« cpw 4598  βˆͺ cuni 4903   class class class wbr 5142  ran crn 5673  βŸΆwf 6538  β€˜cfv 6542  Refcref 23393  LocFinclocfin 23395  Paracompcpcmp 33392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607  ax-inf2 9656  ax-ac2 10478
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8956  df-dom 8957  df-fin 8959  df-r1 9779  df-rank 9780  df-card 9954  df-ac 10131  df-top 22783  df-topon 22800  df-ref 23396  df-locfin 23398  df-cref 33380  df-pcmp 33393
This theorem is referenced by: (None)
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