Theorem ispnrm 23662

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 6885	. . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2		oveq1 7427	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑m ℕ) = (𝐽 ↑m ℕ))
3	2	mpteq1d 5194	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
4	3	rneqd 5919	. . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
5	1, 4	sseq12d 3941	. 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))
6		df-pnrm 23642	. 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}
7	5, 6	elrab2 3648	1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895  ∩ cint 4906   ↦ cmpt 5185  ran crn 5651  ‘cfv 6538  (class class class)co 7420   ↑_m cmap 8850  ℕcn 12338  Clsdccld 23339  Nrmcnrm 23633  PNrmcpnrm 23635

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-cnv 5658  df-dm 5660  df-rn 5661  df-iota 6494  df-fv 6546  df-ov 7423  df-pnrm 23642

This theorem is used by:  pnrmnrm  23663  pnrmcld  23665