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Mirrors > Home > MPE Home > Th. List > lsw | Structured version Visualization version GIF version |
Description: Extract the last symbol of a word. May be not meaningful for other sets which are not words. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
Ref | Expression |
---|---|
lsw | ⊢ (𝑊 ∈ 𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3493 | . 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
2 | fvex 6902 | . 2 ⊢ (𝑊‘((♯‘𝑊) − 1)) ∈ V | |
3 | id 22 | . . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
4 | fveq2 6889 | . . . . 5 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | |
5 | 4 | oveq1d 7421 | . . . 4 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) − 1) = ((♯‘𝑊) − 1)) |
6 | 3, 5 | fveq12d 6896 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑤‘((♯‘𝑤) − 1)) = (𝑊‘((♯‘𝑊) − 1))) |
7 | df-lsw 14510 | . . 3 ⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1))) | |
8 | 6, 7 | fvmptg 6994 | . 2 ⊢ ((𝑊 ∈ V ∧ (𝑊‘((♯‘𝑊) − 1)) ∈ V) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
9 | 1, 2, 8 | sylancl 587 | 1 ⊢ (𝑊 ∈ 𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ‘cfv 6541 (class class class)co 7406 1c1 11108 − cmin 11441 ♯chash 14287 lastSclsw 14509 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6493 df-fun 6543 df-fv 6549 df-ov 7409 df-lsw 14510 |
This theorem is referenced by: lsw0 14512 lsw1 14514 lswcl 14515 ccatval1lsw 14531 lswccatn0lsw 14538 swrdlsw 14614 pfxfvlsw 14642 repswlsw 14729 lswcshw 14762 lswco 14787 lsws2 14852 lsws3 14853 lsws4 14854 wrdl2exs2 14894 swrd2lsw 14900 psgnunilem5 19357 wlkonwlk1l 28910 wwlknlsw 29091 wwlksnext 29137 wwlksnredwwlkn 29139 wwlksnextproplem2 29154 clwlkclwwlklem2a1 29235 clwlkclwwlklem2a3 29237 clwlkclwwlklem2a4 29240 clwlkclwwlklem2 29243 clwwisshclwwslem 29257 clwwlknlbonbgr1 29282 clwwlkn2 29287 clwwlkel 29289 clwwlkf 29290 clwwlkwwlksb 29297 clwwlknonex2lem2 29351 2clwwlk2clwwlklem 29589 numclwwlk1lem2f1 29600 pfxlsw2ccat 32104 iwrdsplit 33375 signsvtn0 33570 signstfveq0 33577 lswn0 46099 |
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