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Theorem ispnrm 22825
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 6888 . . 3 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2 oveq1 7411 . . . . 5 (𝑗 = 𝐽 → (𝑗m ℕ) = (𝐽m ℕ))
32mpteq1d 5242 . . . 4 (𝑗 = 𝐽 → (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
43rneqd 5935 . . 3 (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
51, 4sseq12d 4014 . 2 (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
6 df-pnrm 22805 . 2 PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓)}
75, 6elrab2 3685 1 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wss 3947   cint 4949  cmpt 5230  ran crn 5676  cfv 6540  (class class class)co 7404  m cmap 8816  cn 12208  Clsdccld 22502  Nrmcnrm 22796  PNrmcpnrm 22798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5683  df-dm 5685  df-rn 5686  df-iota 6492  df-fv 6548  df-ov 7407  df-pnrm 22805
This theorem is referenced by:  pnrmnrm  22826  pnrmcld  22828
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