Theorem ispnrm 21654

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 6501	. . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2		oveq1 6985	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑𝑚 ℕ) = (𝐽 ↑𝑚 ℕ))
3	2	mpteq1d 5017	. . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
4	3	rneqd 5652	. . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))
5	1, 4	sseq12d 3892	. 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)))
6		df-pnrm 21634	. 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)}
7	5, 6	elrab2 3599	1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)))

Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ⊆ wss 3831  ∩ cint 4750   ↦ cmpt 5009  ran crn 5409  ‘cfv 6190  (class class class)co 6978   ↑_𝑚 cmap 8208  ℕcn 11441  Clsdccld 21331  Nrmcnrm 21625  PNrmcpnrm 21627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-cnv 5416  df-dm 5418  df-rn 5419  df-iota 6154  df-fv 6198  df-ov 6981  df-pnrm 21634

This theorem is referenced by:  pnrmnrm  21655  pnrmcld  21657