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Mirrors > Home > MPE Home > Th. List > fnmrc | Structured version Visualization version GIF version |
Description: Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
fnmrc | ⊢ mrCls Fn ∪ ran Moore |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mrc 16338 | . . 3 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠})) | |
2 | 1 | fnmpt 6101 | . 2 ⊢ (∀𝑐 ∈ ∪ ran Moore(𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V → mrCls Fn ∪ ran Moore) |
3 | mreunirn 16352 | . . 3 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐)) | |
4 | mrcflem 16357 | . . . . 5 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐) | |
5 | fssxp 6141 | . . . . 5 ⊢ ((𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐 → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐)) |
7 | vuniex 7039 | . . . . . 6 ⊢ ∪ 𝑐 ∈ V | |
8 | 7 | pwex 4921 | . . . . 5 ⊢ 𝒫 ∪ 𝑐 ∈ V |
9 | vex 3275 | . . . . 5 ⊢ 𝑐 ∈ V | |
10 | 8, 9 | xpex 7047 | . . . 4 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V |
11 | ssexg 4880 | . . . 4 ⊢ (((𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ⊆ (𝒫 ∪ 𝑐 × 𝑐) ∧ (𝒫 ∪ 𝑐 × 𝑐) ∈ V) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V) | |
12 | 6, 10, 11 | sylancl 697 | . . 3 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V) |
13 | 3, 12 | sylbi 207 | . 2 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V) |
14 | 2, 13 | mprg 2996 | 1 ⊢ mrCls Fn ∪ ran Moore |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2071 {crab 2986 Vcvv 3272 ⊆ wss 3648 𝒫 cpw 4234 ∪ cuni 4512 ∩ cint 4551 ↦ cmpt 4805 × cxp 5184 ran crn 5187 Fn wfn 5964 ⟶wf 5965 ‘cfv 5969 Moorecmre 16333 mrClscmrc 16334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-ral 2987 df-rex 2988 df-rab 2991 df-v 3274 df-sbc 3510 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-op 4260 df-uni 4513 df-int 4552 df-br 4729 df-opab 4789 df-mpt 4806 df-id 5096 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-fv 5977 df-mre 16337 df-mrc 16338 |
This theorem is referenced by: ismrc 37651 |
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