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Mirrors > Home > MPE Home > Th. List > wrdsymb | Structured version Visualization version GIF version |
Description: A word is a word over the symbols it consists of. (Contributed by AV, 1-Dec-2022.) |
Ref | Expression |
---|---|
wrdsymb | ⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ Word (𝑆 “ (0..^(♯‘𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdf 13864 | . . 3 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴) | |
2 | fimadmfo 6596 | . . 3 ⊢ (𝑆:(0..^(♯‘𝑆))⟶𝐴 → 𝑆:(0..^(♯‘𝑆))–onto→(𝑆 “ (0..^(♯‘𝑆)))) | |
3 | fof 6587 | . . 3 ⊢ (𝑆:(0..^(♯‘𝑆))–onto→(𝑆 “ (0..^(♯‘𝑆))) → 𝑆:(0..^(♯‘𝑆))⟶(𝑆 “ (0..^(♯‘𝑆)))) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶(𝑆 “ (0..^(♯‘𝑆)))) |
5 | iswrdb 13865 | . 2 ⊢ (𝑆 ∈ Word (𝑆 “ (0..^(♯‘𝑆))) ↔ 𝑆:(0..^(♯‘𝑆))⟶(𝑆 “ (0..^(♯‘𝑆)))) | |
6 | 4, 5 | sylibr 236 | 1 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆 ∈ Word (𝑆 “ (0..^(♯‘𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 “ cima 5555 ⟶wf 6348 –onto→wfo 6350 ‘cfv 6352 (class class class)co 7153 0cc0 10534 ..^cfzo 13031 ♯chash 13688 Word cword 13859 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-int 4874 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-1o 8099 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-fin 8510 df-card 9365 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-n0 11896 df-z 11980 df-uz 12242 df-fz 12891 df-fzo 13032 df-hash 13689 df-word 13860 |
This theorem is referenced by: (None) |
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