Theorem ispnrm 23233

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 6861	. . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2		oveq1 7397	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑m ℕ) = (𝐽 ↑m ℕ))
3	2	mpteq1d 5200	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
4	3	rneqd 5905	. . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))
5	1, 4	sseq12d 3983	. 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))
6		df-pnrm 23213	. 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}
7	5, 6	elrab2 3665	1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917  ∩ cint 4913   ↦ cmpt 5191  ran crn 5642  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  ℕcn 12193  Clsdccld 22910  Nrmcnrm 23204  PNrmcpnrm 23206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-cnv 5649  df-dm 5651  df-rn 5652  df-iota 6467  df-fv 6522  df-ov 7393  df-pnrm 23213

This theorem is referenced by:  pnrmnrm  23234  pnrmcld  23236