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| 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 3501 | . 2 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
| 2 | fveq2 6906 | . . . . 5 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | |
| 3 | 2 | oveq2d 7447 | . . . 4 ⊢ (𝑤 = 𝑊 → (0..^(♯‘𝑤)) = (0..^(♯‘𝑊))) |
| 4 | id 22 | . . . . 5 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
| 5 | 2 | oveq1d 7446 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) − 1) = ((♯‘𝑊) − 1)) |
| 6 | 5 | oveq1d 7446 | . . . . 5 ⊢ (𝑤 = 𝑊 → (((♯‘𝑤) − 1) − 𝑥) = (((♯‘𝑊) − 1) − 𝑥)) |
| 7 | 4, 6 | fveq12d 6913 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑤‘(((♯‘𝑤) − 1) − 𝑥)) = (𝑊‘(((♯‘𝑊) − 1) − 𝑥))) |
| 8 | 3, 7 | mpteq12dv 5233 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (0..^(♯‘𝑤)) ↦ (𝑤‘(((♯‘𝑤) − 1) − 𝑥))) = (𝑥 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘(((♯‘𝑊) − 1) − 𝑥)))) |
| 9 | df-reverse 14797 | . . 3 ⊢ reverse = (𝑤 ∈ V ↦ (𝑥 ∈ (0..^(♯‘𝑤)) ↦ (𝑤‘(((♯‘𝑤) − 1) − 𝑥)))) | |
| 10 | ovex 7464 | . . . 4 ⊢ (0..^(♯‘𝑊)) ∈ V | |
| 11 | 10 | mptex 7243 | . . 3 ⊢ (𝑥 ∈ (0..^(♯‘𝑊)) ↦ (𝑊‘(((♯‘𝑊) − 1) − 𝑥))) ∈ V |
| 12 | 8, 9, 11 | fvmpt 7016 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 − cmin 11492 ..^cfzo 13694 ♯chash 14369 reversecreverse 14796 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-reverse 14797 |
| This theorem is referenced by: revcl 14799 revlen 14800 revfv 14801 repswrevw 14825 revco 14873 |
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