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Mirrors > Home > MPE Home > Th. List > wrd2pr2op | Structured version Visualization version GIF version |
Description: A word of length two represented as unordered pair of ordered pairs. (Contributed by AV, 20-Oct-2018.) (Proof shortened by AV, 26-Jan-2021.) |
Ref | Expression |
---|---|
wrd2pr2op | ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = {〈0, (𝑊‘0)〉, 〈1, (𝑊‘1)〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrdfn 13726 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → 𝑊 Fn (0..^(♯‘𝑊))) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 Fn (0..^(♯‘𝑊))) |
3 | oveq2 7031 | . . . . . 6 ⊢ ((♯‘𝑊) = 2 → (0..^(♯‘𝑊)) = (0..^2)) | |
4 | fzo0to2pr 12976 | . . . . . 6 ⊢ (0..^2) = {0, 1} | |
5 | 3, 4 | syl6req 2850 | . . . . 5 ⊢ ((♯‘𝑊) = 2 → {0, 1} = (0..^(♯‘𝑊))) |
6 | 5 | adantl 482 | . . . 4 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → {0, 1} = (0..^(♯‘𝑊))) |
7 | 6 | fneq2d 6324 | . . 3 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → (𝑊 Fn {0, 1} ↔ 𝑊 Fn (0..^(♯‘𝑊)))) |
8 | 2, 7 | mpbird 258 | . 2 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 Fn {0, 1}) |
9 | c0ex 10488 | . . 3 ⊢ 0 ∈ V | |
10 | 1ex 10490 | . . 3 ⊢ 1 ∈ V | |
11 | 9, 10 | fnprb 6844 | . 2 ⊢ (𝑊 Fn {0, 1} ↔ 𝑊 = {〈0, (𝑊‘0)〉, 〈1, (𝑊‘1)〉}) |
12 | 8, 11 | sylib 219 | 1 ⊢ ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 2) → 𝑊 = {〈0, (𝑊‘0)〉, 〈1, (𝑊‘1)〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 {cpr 4480 〈cop 4484 Fn wfn 6227 ‘cfv 6232 (class class class)co 7023 0cc0 10390 1c1 10391 2c2 11546 ..^cfzo 12887 ♯chash 13544 Word cword 13711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-fzo 12888 df-hash 13545 df-word 13712 |
This theorem is referenced by: wrdlen2 14146 wrdlen2s2 14147 |
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