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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 7761 | . . . 4 ⊢ (𝐷 ∈ DirRel → ∪ 𝐷 ∈ V) | |
| 2 | uniexg 7761 | . . . 4 ⊢ (∪ 𝐷 ∈ V → ∪ ∪ 𝐷 ∈ V) | |
| 3 | mptexg 7242 | . . . 4 ⊢ (∪ ∪ 𝐷 ∈ V → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) | |
| 4 | 1, 2, 3 | 3syl 18 | . . 3 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) | 
| 5 | unieq 4917 | . . . . . 6 ⊢ (𝑑 = 𝐷 → ∪ 𝑑 = ∪ 𝐷) | |
| 6 | 5 | unieqd 4919 | . . . . 5 ⊢ (𝑑 = 𝐷 → ∪ ∪ 𝑑 = ∪ ∪ 𝐷) | 
| 7 | imaeq1 6072 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 “ {𝑥}) = (𝐷 “ {𝑥})) | |
| 8 | 6, 7 | mpteq12dv 5232 | . . . 4 ⊢ (𝑑 = 𝐷 → (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥})) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥}))) | 
| 9 | df-tail 18643 | . . . 4 ⊢ tail = (𝑑 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑑 ↦ (𝑑 “ {𝑥}))) | |
| 10 | 8, 9 | fvmptg 7013 | . . 3 ⊢ ((𝐷 ∈ DirRel ∧ (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) ∈ V) → (tail‘𝐷) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥}))) | 
| 11 | 4, 10 | mpdan 687 | . 2 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥}))) | 
| 12 | tailfval.1 | . . . 4 ⊢ 𝑋 = dom 𝐷 | |
| 13 | dirdm 18646 | . . . 4 ⊢ (𝐷 ∈ DirRel → dom 𝐷 = ∪ ∪ 𝐷) | |
| 14 | 12, 13 | eqtr2id 2789 | . . 3 ⊢ (𝐷 ∈ DirRel → ∪ ∪ 𝐷 = 𝑋) | 
| 15 | 14 | mpteq1d 5236 | . 2 ⊢ (𝐷 ∈ DirRel → (𝑥 ∈ ∪ ∪ 𝐷 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) | 
| 16 | 11, 15 | eqtrd 2776 | 1 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 {csn 4625 ∪ cuni 4906 ↦ cmpt 5224 dom cdm 5684 “ cima 5687 ‘cfv 6560 DirRelcdir 18640 tailctail 18641 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-dir 18642 df-tail 18643 | 
| This theorem is referenced by: tailval 36375 tailf 36377 | 
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