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Theorem ispnrm 21948
 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 6649 . . 3 (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽))
2 oveq1 7146 . . . . 5 (𝑗 = 𝐽 → (𝑗m ℕ) = (𝐽m ℕ))
32mpteq1d 5122 . . . 4 (𝑗 = 𝐽 → (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
43rneqd 5776 . . 3 (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) = ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓))
51, 4sseq12d 3951 . 2 (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
6 df-pnrm 21928 . 2 PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗m ℕ) ↦ ran 𝑓)}
75, 6elrab2 3634 1 (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽m ℕ) ↦ ran 𝑓)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ⊆ wss 3884  ∩ cint 4841   ↦ cmpt 5113  ran crn 5524  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  ℕcn 11629  Clsdccld 21625  Nrmcnrm 21919  PNrmcpnrm 21921 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-cnv 5531  df-dm 5533  df-rn 5534  df-iota 6287  df-fv 6336  df-ov 7142  df-pnrm 21928 This theorem is referenced by:  pnrmnrm  21949  pnrmcld  21951
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