| Mathbox for Jeff Hankins |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > tailf | Structured version Visualization version GIF version | ||
| Description: The tail function of a directed set sends its elements to its subsets. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
| Ref | Expression |
|---|---|
| tailf.1 | ⊢ 𝑋 = dom 𝐷 |
| Ref | Expression |
|---|---|
| tailf | ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imassrn 6089 | . . . . . . 7 ⊢ (𝐷 “ {𝑥}) ⊆ ran 𝐷 | |
| 2 | ssun2 4179 | . . . . . . . 8 ⊢ ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷) | |
| 3 | dmrnssfld 5984 | . . . . . . . 8 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷 | |
| 4 | 2, 3 | sstri 3993 | . . . . . . 7 ⊢ ran 𝐷 ⊆ ∪ ∪ 𝐷 |
| 5 | 1, 4 | sstri 3993 | . . . . . 6 ⊢ (𝐷 “ {𝑥}) ⊆ ∪ ∪ 𝐷 |
| 6 | tailf.1 | . . . . . . 7 ⊢ 𝑋 = dom 𝐷 | |
| 7 | dirdm 18645 | . . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷) | |
| 8 | 6, 7 | eqtr2id 2790 | . . . . . 6 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋) |
| 9 | 5, 8 | sseqtrid 4026 | . . . . 5 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋) |
| 10 | dmexg 7923 | . . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 ∈ V) | |
| 11 | 6, 10 | eqeltrid 2845 | . . . . . 6 ⊢ (𝐷 ∈ DirRel → 𝑋 ∈ V) |
| 12 | elpw2g 5333 | . . . . . 6 ⊢ (𝑋 ∈ V → ((𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝐷 “ {𝑥}) ⊆ 𝑋)) | |
| 13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝐷 ∈ DirRel → ((𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝐷 “ {𝑥}) ⊆ 𝑋)) |
| 14 | 9, 13 | mpbird 257 | . . . 4 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋) |
| 15 | 14 | ralrimivw 3150 | . . 3 ⊢ (𝐷 ∈ DirRel → ∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋) |
| 16 | eqid 2737 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) | |
| 17 | 16 | fmpt 7130 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋) |
| 18 | 15, 17 | sylib 218 | . 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋) |
| 19 | 6 | tailfval 36373 | . . 3 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
| 20 | 19 | feq1d 6720 | . 2 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)) |
| 21 | 18, 20 | mpbird 257 | 1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 “ cima 5688 ⟶wf 6557 ‘cfv 6561 DirRelcdir 18639 tailctail 18640 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 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-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-iun 4993 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-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-dir 18641 df-tail 18642 |
| This theorem is referenced by: tailfb 36378 filnetlem4 36382 |
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