Theorem pnrmcld 21146

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 21143	. . . 4 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)))
2	1	simprbi 480	. . 3 ⊢ (𝐽 ∈ PNrm → (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
3	2	sselda 3603	. 2 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → 𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
4		eqid 2622	. . . 4 ⊢ (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)
5	4	elrnmpt 5372	. . 3 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (𝐴 ∈ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) ↔ ∃𝑓 ∈ (𝐽 ↑𝑚 ℕ)𝐴 = ∩ ran 𝑓))
6	5	adantl 482	. 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 222	1 ⊢ ((𝐽 ∈ PNrm ∧ 𝐴 ∈ (Clsd‘𝐽)) → ∃𝑓 ∈ (𝐽 ↑𝑚 ℕ)𝐴 = ∩ ran 𝑓)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ⊆ wss 3574  ∩ cint 4475   ↦ cmpt 4729  ran crn 5115  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  ℕcn 11020  Clsdccld 20820  Nrmcnrm 21114  PNrmcpnrm 21116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-ov 6653  df-pnrm 21123

This theorem is referenced by:  pnrmopn  21147