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Mirrors > Home > MPE Home > Th. List > Mathboxes > tailval | Structured version Visualization version GIF version |
Description: The tail of an element in 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 |
---|---|
tailval | ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tailfval.1 | . . . . 5 ⊢ 𝑋 = dom 𝐷 | |
2 | 1 | tailfval 32694 | . . . 4 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
3 | 2 | fveq1d 6355 | . . 3 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴)) |
4 | 3 | adantr 472 | . 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴)) |
5 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋) | |
6 | imaexg 7269 | . . 3 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V) | |
7 | sneq 4331 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
8 | 7 | imaeq2d 5624 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴})) |
9 | eqid 2760 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) | |
10 | 8, 9 | fvmptg 6443 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴})) |
11 | 5, 6, 10 | syl2anr 496 | . 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴})) |
12 | 4, 11 | eqtrd 2794 | 1 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 {csn 4321 ↦ cmpt 4881 dom cdm 5266 “ cima 5269 ‘cfv 6049 DirRelcdir 17449 tailctail 17450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pr 5055 ax-un 7115 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-dir 17451 df-tail 17452 |
This theorem is referenced by: eltail 32696 |
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