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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 6670 | . . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽)) | |
2 | oveq1 7163 | . . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑m ℕ) = (𝐽 ↑m ℕ)) | |
3 | 2 | mpteq1d 5155 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)) |
4 | 3 | rneqd 5808 | . . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓)) |
5 | 1, 4 | sseq12d 4000 | . 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))) |
6 | df-pnrm 21927 | . 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)} | |
7 | 5, 6 | elrab2 3683 | 1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑m ℕ) ↦ ∩ ran 𝑓))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∩ cint 4876 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 ℕcn 11638 Clsdccld 21624 Nrmcnrm 21918 PNrmcpnrm 21920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-cnv 5563 df-dm 5565 df-rn 5566 df-iota 6314 df-fv 6363 df-ov 7159 df-pnrm 21927 |
This theorem is referenced by: pnrmnrm 21948 pnrmcld 21950 |
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