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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 22418 for topologies. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
fnmre | ⊢ Moore Fn V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vpwex 5375 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
2 | 1 | pwex 5378 | . . 3 ⊢ 𝒫 𝒫 𝑥 ∈ V |
3 | 2 | rabex 5332 | . 2 ⊢ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))} ∈ V |
4 | df-mre 17527 | . 2 ⊢ Moore = (𝑥 ∈ V ↦ {𝑐 ∈ 𝒫 𝒫 𝑥 ∣ (𝑥 ∈ 𝑐 ∧ ∀𝑠 ∈ 𝒫 𝑐(𝑠 ≠ ∅ → ∩ 𝑠 ∈ 𝑐))}) | |
5 | 3, 4 | fnmpti 6691 | 1 ⊢ Moore Fn V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 {crab 3433 Vcvv 3475 ∅c0 4322 𝒫 cpw 4602 ∩ cint 4950 Fn wfn 6536 Moorecmre 17523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-fun 6543 df-fn 6544 df-mre 17527 |
This theorem is referenced by: mreunirn 17542 |
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