Theorem pcmplfinf 34282

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 1232	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑈 ⊆ 𝐽)
3		simpll3 1233	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑋 = ∪ 𝑈)
4		elpwi 4574	. . . 4 ⊢ (𝑣 ∈ 𝒫 𝐽 → 𝑣 ⊆ 𝐽)
5	4	ad2antlr 740	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣 ⊆ 𝐽)
6		simprr 785	. . 3 ⊢ ((((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) ∧ 𝑣 ∈ 𝒫 𝐽) ∧ (𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈)) → 𝑣Ref𝑈)
7		simprl 783	. . 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 34262	. 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 34281	. 2 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑣 ∈ 𝒫 𝐽(𝑣 ∈ (LocFin‘𝐽) ∧ 𝑣Ref𝑈))
10	8, 9	r19.29a 3176	1 ⊢ ((𝐽 ∈ Paracomp ∧ 𝑈 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑈) → ∃𝑓(𝑓:𝑈⟶𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877   class class class wbr 5114  ran crn 5667  ⟶wf 6539  ‘cfv 6543  Refcref 23696  LocFinclocfin 23698  Paracompcpcmp 34276

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-reg 9564  ax-inf2 9620  ax-ac2 10465

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7380  df-ov 7426  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-fin 8956  df-r1 9746  df-rank 9747  df-scott 9868  df-card 9944  df-ac 10119  df-top 23088  df-topon 23105  df-ref 23699  df-locfin 23701  df-cref 34264  df-pcmp 34277

This theorem is used by: (None)