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Theorem ispnrm 22842
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 6891 . . 3 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2 oveq1 7415 . . . . 5 (𝑗 = 𝐽 → (𝑗m ℕ) = (𝐽m ℕ))
32mpteq1d 5243 . . . 4 (𝑗 = 𝐽 → (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
43rneqd 5937 . . 3 (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
51, 4sseq12d 4015 . 2 (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
6 df-pnrm 22822 . 2 PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓)}
75, 6elrab2 3686 1 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  wss 3948   cint 4950  cmpt 5231  ran crn 5677  cfv 6543  (class class class)co 7408  m cmap 8819  cn 12211  Clsdccld 22519  Nrmcnrm 22813  PNrmcpnrm 22815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551  df-ov 7411  df-pnrm 22822
This theorem is referenced by:  pnrmnrm  22843  pnrmcld  22845
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