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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 34206 | . . . 4 ⊢ (𝐷 ∈ DirRel → (tail‘𝐷) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))) |
3 | 2 | fveq1d 6678 | . . 3 ⊢ (𝐷 ∈ DirRel → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴)) |
4 | 3 | adantr 484 | . 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴)) |
5 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋) | |
6 | imaexg 7648 | . . 3 ⊢ (𝐷 ∈ DirRel → (𝐷 “ {𝐴}) ∈ V) | |
7 | sneq 4526 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
8 | 7 | imaeq2d 5903 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐷 “ {𝑥}) = (𝐷 “ {𝐴})) |
9 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) = (𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥})) | |
10 | 8, 9 | fvmptg 6775 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ (𝐷 “ {𝐴}) ∈ V) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴})) |
11 | 5, 6, 10 | syl2anr 600 | . 2 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ (𝐷 “ {𝑥}))‘𝐴) = (𝐷 “ {𝐴})) |
12 | 4, 11 | eqtrd 2773 | 1 ⊢ ((𝐷 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → ((tail‘𝐷)‘𝐴) = (𝐷 “ {𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 {csn 4516 ↦ cmpt 5110 dom cdm 5525 “ cima 5528 ‘cfv 6339 DirRelcdir 17956 tailctail 17957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7481 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-dir 17958 df-tail 17959 |
This theorem is referenced by: eltail 34208 |
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