Theorem pnrmcld 23371

Description: A closed set in a perfectly normal space is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
pnrmcld	⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓)

Distinct variable groups:   𝐴,𝑓   𝑓,𝐽

Proof of Theorem pnrmcld

Step	Hyp	Ref	Expression
1		ispnrm 23368	. . . 4 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))
2	1	simprbi 496	. . 3 ⊢ (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
3	2	sselda 4008	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
4		eqid 2740	. . . 4 ⊢ (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)
5	4	elrnmpt 5981	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓))
6	5	adantl 481	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓))
7	3, 6	mpbid 232	1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽 ↑m ℕ)𝐴 = ∩ ran 𝑓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   ⊆ wss 3976  ∩ cint 4970   ↦ cmpt 5249  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  ℕcn 12293  Clsdccld 23045  Nrmcnrm 23339  PNrmcpnrm 23341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711  df-iota 6525  df-fv 6581  df-ov 7451  df-pnrm 23348

This theorem is referenced by:  pnrmopn  23372