Theorem ispnrm 20953

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 6103	. . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2		oveq1 6556	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑𝑚 ℕ) = (𝐽 ↑𝑚 ℕ))
3	2	mpteq1d 4666	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
4	3	rneqd 5274	. . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
5	1, 4	sseq12d 3597	. 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)))
6		df-pnrm 20933	. 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)}
7	5, 6	elrab2 3333	1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∩ cint 4410   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  ℕcn 10897  Clsdccld 20630  Nrmcnrm 20924  PNrmcpnrm 20926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812  df-ov 6552  df-pnrm 20933

This theorem is referenced by:  pnrmnrm  20954  pnrmcld  20956