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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 5940 | . . . . . . 7 ⊢ (𝐷 “ {𝑥}) ⊆ ran 𝐷 | |
2 | ssun2 4149 | . . . . . . . 8 ⊢ ran 𝐷 ⊆ (dom 𝐷 ∪ ran 𝐷) | |
3 | dmrnssfld 5841 | . . . . . . . 8 ⊢ (dom 𝐷 ∪ ran 𝐷) ⊆ ∪ ∪ 𝐷 | |
4 | 2, 3 | sstri 3976 | . . . . . . 7 ⊢ ran 𝐷 ⊆ ∪ ∪ 𝐷 |
5 | 1, 4 | sstri 3976 | . . . . . 6 ⊢ (𝐷 “ {𝑥}) ⊆ ∪ ∪ 𝐷 |
6 | tailf.1 | . . . . . . 7 ⊢ 𝑋 = dom 𝐷 | |
7 | dirdm 17844 | . . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷) | |
8 | 6, 7 | syl5req 2869 | . . . . . 6 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋) |
9 | 5, 8 | sseqtrid 4019 | . . . . 5 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ⊆ 𝑋) |
10 | dmexg 7613 | . . . . . . 7 ⊢ (𝐷 ∈ DirRel → dom 𝐷 ∈ V) | |
11 | 6, 10 | eqeltrid 2917 | . . . . . 6 ⊢ (𝐷 ∈ DirRel → 𝑋 ∈ V) |
12 | elpw2g 5247 | . . . . . 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 259 | . . . 4 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝑥}) ∈ 𝒫 𝑋) |
15 | 14 | ralrimivw 3183 | . . 3 ⊢ (𝐷 ∈ DirRel → ∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋) |
16 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) | |
17 | 16 | fmpt 6874 | . . 3 ⊢ (∀𝑥 ∈ 𝑋 (𝐷 “ {𝑥}) ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋) |
18 | 15, 17 | sylib 220 | . 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋) |
19 | 6 | tailfval 33720 | . . 3 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
20 | 19 | feq1d 6499 | . 2 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷):𝑋⟶𝒫 𝑋 ↔ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})):𝑋⟶𝒫 𝑋)) |
21 | 18, 20 | mpbird 259 | 1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷):𝑋⟶𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ∪ cun 3934 ⊆ wss 3936 𝒫 cpw 4539 {csn 4567 ∪ cuni 4838 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 “ cima 5558 ⟶wf 6351 ‘cfv 6355 DirRelcdir 17838 tailctail 17839 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-dir 17840 df-tail 17841 |
This theorem is referenced by: tailfb 33725 filnetlem4 33729 |
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