![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > revval | Structured version Visualization version GIF version |
Description: Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
Ref | Expression |
---|---|
revval | ⊢ (𝑊 ∈ 𝑉 → (reverse‘𝑊) = (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3243 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | fveq2 6229 | . . . . 5 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊)) | |
3 | 2 | oveq2d 6706 | . . . 4 ⊢ (𝑤 = 𝑊 → (0..^(#‘𝑤)) = (0..^(#‘𝑊))) |
4 | id 22 | . . . . 5 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
5 | 2 | oveq1d 6705 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ((#‘𝑤) − 1) = ((#‘𝑊) − 1)) |
6 | 5 | oveq1d 6705 | . . . . 5 ⊢ (𝑤 = 𝑊 → (((#‘𝑤) − 1) − 𝑥) = (((#‘𝑊) − 1) − 𝑥)) |
7 | 4, 6 | fveq12d 6235 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤‘(((#‘𝑤) − 1) − 𝑥)) = (𝑊‘(((#‘𝑊) − 1) − 𝑥))) |
8 | 3, 7 | mpteq12dv 4766 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (0..^(#‘𝑤)) ↦ (𝑤‘(((#‘𝑤) − 1) − 𝑥))) = (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥)))) |
9 | df-reverse 13337 | . . 3 ⊢ reverse = (𝑤 ∈ V ↦ (𝑥 ∈ (0..^(#‘𝑤)) ↦ (𝑤‘(((#‘𝑤) − 1) − 𝑥)))) | |
10 | ovex 6718 | . . . 4 ⊢ (0..^(#‘𝑊)) ∈ V | |
11 | 10 | mptex 6527 | . . 3 ⊢ (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥))) ∈ V |
12 | 8, 9, 11 | fvmpt 6321 | . 2 ⊢ (𝑊 ∈ V → (reverse‘𝑊) = (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥)))) |
13 | 1, 12 | syl 17 | 1 ⊢ (𝑊 ∈ 𝑉 → (reverse‘𝑊) = (𝑥 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘(((#‘𝑊) − 1) − 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 0cc0 9974 1c1 9975 − cmin 10304 ..^cfzo 12504 #chash 13157 reversecreverse 13329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-reverse 13337 |
This theorem is referenced by: revcl 13556 revlen 13557 revfv 13558 repswrevw 13579 revco 13626 |
Copyright terms: Public domain | W3C validator |