Theorem pcmplfinf 31856

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 ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ 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.

Step	Hyp	Ref	Expression
1		pcmplfin.x	. . 3 ⊢ 𝑋 = ∪ 𝐽
2		simpll2 1213	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑈 ⊆ 𝐽)
3		simpll3 1214	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑋 = ∪ 𝑈)
4		elpwi 4546	. . . 4 ⊢ (𝑣 ∈ 𝒫 𝐽 → 𝑣 ⊆ 𝐽)
5	4	ad2antlr 725	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣 ⊆ 𝐽)
6		simprr 771	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣Ref𝑈)
7		simprl 769	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣 ∈ (LocFin‘𝐽))
8	1, 2, 3, 5, 6, 7	locfinref 31836	. 2 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
9	1	pcmplfin 31855	. 2 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
10	8, 9	r19.29a 3156	1 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087   = wceq 1539  ∃wex 1779   ∈ wcel 2104   ⊆ wss 3892  𝒫 cpw 4539  ∪ cuni 4844   class class class wbr 5081  ran crn 5601  ⟶wf 6454  ‘cfv 6458  Refcref 22698  LocFinclocfin 22700  Paracompcpcmp 31850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-reg 9395  ax-inf2 9443  ax-ac2 10265

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-iin 4934  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-er 8529  df-en 8765  df-dom 8766  df-fin 8768  df-r1 9566  df-rank 9567  df-card 9741  df-ac 9918  df-top 22088  df-topon 22105  df-ref 22701  df-locfin 22703  df-cref 31838  df-pcmp 31851

This theorem is referenced by: (None)