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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 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))
Distinct variable group:   𝑓,𝐽

Proof of Theorem ispnrm
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6353 . . 3 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2 oveq1 6821 . . . . 5 (𝑗 = 𝐽 → (𝑗𝑚 ℕ) = (𝐽𝑚 ℕ))
32mpteq1d 4890 . . . 4 (𝑗 = 𝐽 → (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
43rneqd 5508 . . 3 (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓) = ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓))
51, 4sseq12d 3775 . 2 (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))
6 df-pnrm 21345 . 2 PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)}
75, 6elrab2 3507 1 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽𝑚 ℕ) ↦ ran 𝑓)))
Colors of variables: wff setvar class
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  cn 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
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