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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 17630 | . . 3 ⊢ mrCls = (𝑐 ∈ ∪ ran Moore ↦ (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠})) | |
| 2 | 1 | fnmpt 6708 | . 2 ⊢ (∀𝑐 ∈ ∪ ran Moore(𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V → mrCls Fn ∪ ran Moore) |
| 3 | mreunirn 17644 | . . 3 ⊢ (𝑐 ∈ ∪ ran Moore ↔ 𝑐 ∈ (Moore‘∪ 𝑐)) | |
| 4 | mrcflem 17649 | . . . . 5 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}):𝒫 ∪ 𝑐⟶𝑐) | |
| 5 | fssxp 6763 | . . . . 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 7759 | . . . . . 6 ⊢ ∪ 𝑐 ∈ V | |
| 8 | 7 | pwex 5380 | . . . . 5 ⊢ 𝒫 ∪ 𝑐 ∈ V |
| 9 | vex 3484 | . . . . 5 ⊢ 𝑐 ∈ V | |
| 10 | 8, 9 | xpex 7773 | . . . 4 ⊢ (𝒫 ∪ 𝑐 × 𝑐) ∈ V |
| 11 | ssexg 5323 | . . . 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 586 | . . 3 ⊢ (𝑐 ∈ (Moore‘∪ 𝑐) → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V) |
| 13 | 3, 12 | sylbi 217 | . 2 ⊢ (𝑐 ∈ ∪ ran Moore → (𝑥 ∈ 𝒫 ∪ 𝑐 ↦ ∩ {𝑠 ∈ 𝑐 ∣ 𝑥 ⊆ 𝑠}) ∈ V) |
| 14 | 2, 13 | mprg 3067 | 1 ⊢ mrCls Fn ∪ ran Moore |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ∩ cint 4946 ↦ cmpt 5225 × cxp 5683 ran crn 5686 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 Moorecmre 17625 mrClscmrc 17626 |
| 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 ax-un 7755 |
| 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-ne 2941 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-uni 4908 df-int 4947 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-mre 17629 df-mrc 17630 |
| This theorem is referenced by: ismrc 42712 |
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