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| Mirrors > Home > MPE Home > Th. List > fnmre | Structured version Visualization version GIF version | ||
| Description: The Moore collection generator is a well-behaved function. Analogue for Moore collections of fntopon 22930 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
| Ref | Expression |
|---|---|
| fnmre | ⊢ Moore Fn V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vpwex 5377 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
| 2 | 1 | pwex 5380 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
| 3 | 2 | rabex 5339 | . 2 ⊢ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} ∈ V |
| 4 | df-mre 17629 | . 2 ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) | |
| 5 | 3, 4 | fnmpti 6711 | 1 ⊢ Moore Fn V |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 {crab 3436 Vcvv 3480 ∅c0 4333 𝒫 cpw 4600 ∩ cint 4946 Fn wfn 6556 Moorecmre 17625 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 df-mre 17629 |
| This theorem is referenced by: mreunirn 17644 |
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