Theorem ispnrm 22398

Description: The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.)

Assertion

Ref	Expression
ispnrm	⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))

Distinct variable group:   𝑓,𝐽

Proof of Theorem ispnrm

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6756	. . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2		oveq1 7262	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑m ℕ) = (𝐽 ↑m ℕ))
3	2	mpteq1d 5165	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
4	3	rneqd 5836	. . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
5	1, 4	sseq12d 3950	. 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))
6		df-pnrm 22378	. 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}
7	5, 6	elrab2 3620	1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ∩ cint 4876   ↦ cmpt 5153  ran crn 5581  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  ℕcn 11903  Clsdccld 22075  Nrmcnrm 22369  PNrmcpnrm 22371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-cnv 5588  df-dm 5590  df-rn 5591  df-iota 6376  df-fv 6426  df-ov 7258  df-pnrm 22378

This theorem is referenced by:  pnrmnrm  22399  pnrmcld  22401