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| Mirrors > Home > MPE Home > Th. List > mrcssv | Structured version Visualization version GIF version | ||
| Description: The closure of a set is a subset of the base. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
| Ref | Expression |
|---|---|
| mrcfval.f | ⊢ 𝐹 = (mrCls‘𝐶) |
| Ref | Expression |
|---|---|
| mrcssv | ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvssunirn 6939 | . 2 ⊢ (𝐹‘𝑈) ⊆ ∪ ran 𝐹 | |
| 2 | mrcfval.f | . . . . 5 ⊢ 𝐹 = (mrCls‘𝐶) | |
| 3 | 2 | mrcf 17652 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶) |
| 4 | frn 6743 | . . . 4 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶) | |
| 5 | uniss 4915 | . . . 4 ⊢ (ran 𝐹 ⊆ 𝐶 → ∪ ran 𝐹 ⊆ ∪ 𝐶) | |
| 6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ ∪ 𝐶) |
| 7 | mreuni 17643 | . . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ 𝐶 = 𝑋) | |
| 8 | 6, 7 | sseqtrd 4020 | . 2 ⊢ (𝐶 ∈ (Moore‘𝑋) → ∪ ran 𝐹 ⊆ 𝑋) |
| 9 | 1, 8 | sstrid 3995 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐹‘𝑈) ⊆ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ran crn 5686 ⟶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: mrcidb 17658 mrcuni 17664 mrcssvd 17666 mrefg2 42718 proot1hash 43207 |
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