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Mathbox for Jeff Hankins |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tailfval | Structured version Visualization version GIF version |
Description: The tail function for a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 24-Nov-2013.) |
Ref | Expression |
---|---|
tailfval.1 | ⊢ 𝑋 = dom 𝐷 |
Ref | Expression |
---|---|
tailfval | ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 7751 | . . . 4 ⊢ (𝐷 ∈ DirRel → ∪ 𝐷 ∈ V) | |
2 | uniexg 7751 | . . . 4 ⊢ (∪ 𝐷 ∈ V → ∪ ∪ 𝐷 ∈ V) | |
3 | mptexg 7238 | . . . 4 ⊢ (∪ ∪ 𝐷 ∈ V → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) | |
4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) |
5 | unieq 4924 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ∪ 𝑑 = ∪ 𝐷) | |
6 | 5 | unieqd 4926 | . . . . 5 ⊢ (𝑑 = 𝐷 → ∪ ∪ 𝑑 = ∪ ∪ 𝐷) |
7 | imaeq1 6064 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥})) | |
8 | 6, 7 | mpteq12dv 5244 | . . . 4 ⊢ (𝑑 = 𝐷 → (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥}))) |
9 | df-tail 18622 | . . . 4 ⊢ tail = (𝑑 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥}))) | |
10 | 8, 9 | fvmptg 7007 | . . 3 ⊢ ((𝐷 ∈ DirRel ∧ (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥}))) |
11 | 4, 10 | mpdan 685 | . 2 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥}))) |
12 | tailfval.1 | . . . 4 ⊢ 𝑋 = dom 𝐷 | |
13 | dirdm 18625 | . . . 4 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷) | |
14 | 12, 13 | eqtr2id 2779 | . . 3 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋) |
15 | 14 | mpteq1d 5248 | . 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
16 | 11, 15 | eqtrd 2766 | 1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 {csn 4633 ∪ cuni 4913 ↦ cmpt 5236 dom cdm 5682 “ cima 5685 ‘cfv 6554 DirRelcdir 18619 tailctail 18620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-dir 18621 df-tail 18622 |
This theorem is referenced by: tailval 36085 tailf 36087 |
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