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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 3488 | . 2 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V) | |
2 | fvex 6888 | . 2 ⊢ (𝑊‘((♯‘𝑊) − 1)) ∈ V | |
3 | id 22 | . . . 4 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
4 | fveq2 6875 | . . . . 5 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | |
5 | 4 | oveq1d 7405 | . . . 4 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) − 1) = ((♯‘𝑊) − 1)) |
6 | 3, 5 | fveq12d 6882 | . . 3 ⊢ (𝑤 = 𝑊 → (𝑤‘((♯‘𝑤) − 1)) = (𝑊‘((♯‘𝑊) − 1))) |
7 | df-lsw 14492 | . . 3 ⊢ lastS = (𝑤 ∈ V ↦ (𝑤‘((♯‘𝑤) − 1))) | |
8 | 6, 7 | fvmptg 6979 | . 2 ⊢ ((𝑊 ∈ V ∧ (𝑊‘((♯‘𝑊) − 1)) ∈ V) → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
9 | 1, 2, 8 | sylancl 586 | 1 ⊢ (𝑊 ∈ 𝑋 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3470 ‘cfv 6529 (class class class)co 7390 1c1 11090 − cmin 11423 ♯chash 14269 lastSclsw 14491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6481 df-fun 6531 df-fv 6537 df-ov 7393 df-lsw 14492 |
This theorem is referenced by: lsw0 14494 lsw1 14496 lswcl 14497 ccatval1lsw 14513 lswccatn0lsw 14520 swrdlsw 14596 pfxfvlsw 14624 repswlsw 14711 lswcshw 14744 lswco 14769 lsws2 14834 lsws3 14835 lsws4 14836 wrdl2exs2 14876 swrd2lsw 14882 psgnunilem5 19323 wlkonwlk1l 28780 wwlknlsw 28961 wwlksnext 29007 wwlksnredwwlkn 29009 wwlksnextproplem2 29024 clwlkclwwlklem2a1 29105 clwlkclwwlklem2a3 29107 clwlkclwwlklem2a4 29110 clwlkclwwlklem2 29113 clwwisshclwwslem 29127 clwwlknlbonbgr1 29152 clwwlkn2 29157 clwwlkel 29159 clwwlkf 29160 clwwlkwwlksb 29167 clwwlknonex2lem2 29221 2clwwlk2clwwlklem 29459 numclwwlk1lem2f1 29470 pfxlsw2ccat 31982 iwrdsplit 33201 signsvtn0 33396 signstfveq0 33403 lswn0 45870 |
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