![]() |
Mathbox for Jeff Hankins |
< Previous
Next >
Nearby theorems |
|
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 6090 | . . . . . . 7 ⊢ (𝐷 “ {𝑥}) ⊆ ran 𝐷 | |
2 | ssun2 4188 | . . . . . . . 8 ⊢ ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷) | |
3 | dmrnssfld 5986 | . . . . . . . 8 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷 | |
4 | 2, 3 | sstri 4004 | . . . . . . 7 ⊢ ran 𝐷 ⊆ ∪ ∪ 𝐷 |
5 | 1, 4 | sstri 4004 | . . . . . 6 ⊢ (𝐷 “ {𝑥}) ⊆ ∪ ∪ 𝐷 |
6 | tailf.1 | . . . . . . 7 ⊢ 𝑋 = dom 𝐷 | |
7 | dirdm 18657 | . . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷) | |
8 | 6, 7 | eqtr2id 2787 | . . . . . 6 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋) |
9 | 5, 8 | sseqtrid 4047 | . . . . 5 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋) |
10 | dmexg 7923 | . . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 ∈ V) | |
11 | 6, 10 | eqeltrid 2842 | . . . . . 6 ⊢ (𝐷 ∈ DirRel → 𝑋 ∈ V) |
12 | elpw2g 5338 | . . . . . 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 3147 | . . 3 ⊢ (𝐷 ∈ DirRel → ∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋) |
16 | eqid 2734 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) | |
17 | 16 | fmpt 7129 | . . 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 36354 | . . 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 1536 ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ∪ cun 3960 ⊆ wss 3962 𝒫 cpw 4604 {csn 4630 ∪ cuni 4911 ↦ cmpt 5230 dom cdm 5688 ran crn 5689 “ cima 5691 ⟶wf 6558 ‘cfv 6562 DirRelcdir 18651 tailctail 18652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-dir 18653 df-tail 18654 |
This theorem is referenced by: tailfb 36359 filnetlem4 36363 |
Copyright terms: Public domain | W3C validator |