Theorem ispnrm 21365

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 6353	. . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2		oveq1 6821	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑𝑚 ℕ) = (𝐽 ↑𝑚 ℕ))
3	2	mpteq1d 4890	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
4	3	rneqd 5508	. . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
5	1, 4	sseq12d 3775	. 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)))
6		df-pnrm 21345	. 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)}
7	5, 6	elrab2 3507	1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)))

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ⊆ wss 3715  ∩ cint 4627   ↦ cmpt 4881  ran crn 5267  ‘cfv 6049  (class class class)co 6814   ↑_𝑚 cmap 8025  ℕcn 11232  Clsdccld 21042  Nrmcnrm 21336  PNrmcpnrm 21338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-cnv 5274  df-dm 5276  df-rn 5277  df-iota 6012  df-fv 6057  df-ov 6817  df-pnrm 21345

This theorem is referenced by:  pnrmnrm  21366  pnrmcld  21368