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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 34561 | . . . 4 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
3 | 2 | fveq1d 6776 | . . 3 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴)) |
4 | 3 | adantr 481 | . 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴)) |
5 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋) | |
6 | imaexg 7762 | . . 3 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V) | |
7 | sneq 4571 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
8 | 7 | imaeq2d 5969 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴})) |
9 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) | |
10 | 8, 9 | fvmptg 6873 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴})) |
11 | 5, 6, 10 | syl2anr 597 | . 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴})) |
12 | 4, 11 | eqtrd 2778 | 1 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 {csn 4561 ↦ cmpt 5157 dom cdm 5589 “ cima 5592 ‘cfv 6433 DirRelcdir 18312 tailctail 18313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-dir 18314 df-tail 18315 |
This theorem is referenced by: eltail 34563 |
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