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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 6191 | . . 3 ⊢ (𝑗 = 𝐽 → (Clsd‘𝑗) = (Clsd‘𝐽)) | |
2 | oveq1 6657 | . . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ↑𝑚 ℕ) = (𝐽 ↑𝑚 ℕ)) | |
3 | 2 | mpteq1d 4738 | . . . 4 ⊢ (𝑗 = 𝐽 → (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)) |
4 | 3 | rneqd 5353 | . . 3 ⊢ (𝑗 = 𝐽 → ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) = ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)) |
5 | 1, 4 | sseq12d 3634 | . 2 ⊢ (𝑗 = 𝐽 → ((Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓) ↔ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))) |
6 | df-pnrm 21123 | . 2 ⊢ PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑𝑚 ℕ) ↦ ∩ ran 𝑓)} | |
7 | 5, 6 | elrab2 3366 | 1 ⊢ (𝐽 ∈ PNrm ↔ (𝐽 ∈ Nrm ∧ (Clsd‘𝐽) ⊆ ran (𝑓 ∈ (𝐽 ↑𝑚 ℕ) ↦ ∩ ran 𝑓))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∩ cint 4475 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 ℕcn 11020 Clsdccld 20820 Nrmcnrm 21114 PNrmcpnrm 21116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-cnv 5122 df-dm 5124 df-rn 5125 df-iota 5851 df-fv 5896 df-ov 6653 df-pnrm 21123 |
This theorem is referenced by: pnrmnrm 21144 pnrmcld 21146 |
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