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Theorem ispnrm 23363
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 6907 . . 3 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2 oveq1 7438 . . . . 5 (𝑗 = 𝐽 → (𝑗m ℕ) = (𝐽m ℕ))
32mpteq1d 5243 . . . 4 (𝑗 = 𝐽 → (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
43rneqd 5952 . . 3 (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
51, 4sseq12d 4029 . 2 (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
6 df-pnrm 23343 . 2 PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓)}
75, 6elrab2 3698 1 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963   cint 4951  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  m cmap 8865  cn 12264  Clsdccld 23040  Nrmcnrm 23334  PNrmcpnrm 23336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700  df-iota 6516  df-fv 6571  df-ov 7434  df-pnrm 23343
This theorem is referenced by:  pnrmnrm  23364  pnrmcld  23366
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