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| Mirrors > Home > MPE Home > Th. List > ispnrm | Structured version Visualization version GIF version | ||
| Description: The property of being perfectly normal. (Contributed by Mario Carneiro, 26-Aug-2015.) |
| Ref | Expression |
|---|---|
| ispnrm | ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽)) | |
| 2 | oveq1 7438 | . . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑m ℕ) = (𝐽 ↑m ℕ)) | |
| 3 | 2 | mpteq1d 5237 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)) |
| 4 | 3 | rneqd 5949 | . . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)) |
| 5 | 1, 4 | sseq12d 4017 | . 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))) |
| 6 | df-pnrm 23327 | . 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)} | |
| 7 | 5, 6 | elrab2 3695 | 1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∩ cint 4946 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℕcn 12266 Clsdccld 23024 Nrmcnrm 23318 PNrmcpnrm 23320 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 df-ov 7434 df-pnrm 23327 |
| This theorem is referenced by: pnrmnrm 23348 pnrmcld 23350 |
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