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Mirrors > Home > MPE Home > Th. List > wwlktovf | Structured version Visualization version GIF version |
Description: Lemma 1 for wrd2f1tovbij 14323. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
Ref | Expression |
---|---|
wrd2f1tovbij.d | ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} |
wrd2f1tovbij.r | ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} |
wrd2f1tovbij.f | ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) |
Ref | Expression |
---|---|
wwlktovf | ⊢ 𝐹:𝐷⟶𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrd2f1tovbij.f | . 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) | |
2 | wrdf 13865 | . . . . 5 ⊢ (𝑡 ∈ Word 𝑉 → 𝑡:(0..^(♯‘𝑡))⟶𝑉) | |
3 | oveq2 7163 | . . . . . . . 8 ⊢ ((♯‘𝑡) = 2 → (0..^(♯‘𝑡)) = (0..^2)) | |
4 | 3 | feq2d 6499 | . . . . . . 7 ⊢ ((♯‘𝑡) = 2 → (𝑡:(0..^(♯‘𝑡))⟶𝑉 ↔ 𝑡:(0..^2)⟶𝑉)) |
5 | 1nn0 11912 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
6 | 2nn 11709 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
7 | 1lt2 11807 | . . . . . . . . 9 ⊢ 1 < 2 | |
8 | elfzo0 13077 | . . . . . . . . 9 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2)) | |
9 | 5, 6, 7, 8 | mpbir3an 1337 | . . . . . . . 8 ⊢ 1 ∈ (0..^2) |
10 | ffvelrn 6848 | . . . . . . . 8 ⊢ ((𝑡:(0..^2)⟶𝑉 ∧ 1 ∈ (0..^2)) → (𝑡‘1) ∈ 𝑉) | |
11 | 9, 10 | mpan2 689 | . . . . . . 7 ⊢ (𝑡:(0..^2)⟶𝑉 → (𝑡‘1) ∈ 𝑉) |
12 | 4, 11 | syl6bi 255 | . . . . . 6 ⊢ ((♯‘𝑡) = 2 → (𝑡:(0..^(♯‘𝑡))⟶𝑉 → (𝑡‘1) ∈ 𝑉)) |
13 | 12 | 3ad2ant1 1129 | . . . . 5 ⊢ (((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → (𝑡:(0..^(♯‘𝑡))⟶𝑉 → (𝑡‘1) ∈ 𝑉)) |
14 | 2, 13 | mpan9 509 | . . . 4 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → (𝑡‘1) ∈ 𝑉) |
15 | preq1 4668 | . . . . . . . 8 ⊢ ((𝑡‘0) = 𝑃 → {(𝑡‘0), (𝑡‘1)} = {𝑃, (𝑡‘1)}) | |
16 | 15 | eleq1d 2897 | . . . . . . 7 ⊢ ((𝑡‘0) = 𝑃 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
17 | 16 | biimpa 479 | . . . . . 6 ⊢ (((𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → {𝑃, (𝑡‘1)} ∈ 𝑋) |
18 | 17 | 3adant1 1126 | . . . . 5 ⊢ (((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → {𝑃, (𝑡‘1)} ∈ 𝑋) |
19 | 18 | adantl 484 | . . . 4 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → {𝑃, (𝑡‘1)} ∈ 𝑋) |
20 | 14, 19 | jca 514 | . . 3 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → ((𝑡‘1) ∈ 𝑉 ∧ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
21 | fveqeq2 6678 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((♯‘𝑤) = 2 ↔ (♯‘𝑡) = 2)) | |
22 | fveq1 6668 | . . . . . 6 ⊢ (𝑤 = 𝑡 → (𝑤‘0) = (𝑡‘0)) | |
23 | 22 | eqeq1d 2823 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((𝑤‘0) = 𝑃 ↔ (𝑡‘0) = 𝑃)) |
24 | fveq1 6668 | . . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤‘1) = (𝑡‘1)) | |
25 | 22, 24 | preq12d 4676 | . . . . . 6 ⊢ (𝑤 = 𝑡 → {(𝑤‘0), (𝑤‘1)} = {(𝑡‘0), (𝑡‘1)}) |
26 | 25 | eleq1d 2897 | . . . . 5 ⊢ (𝑤 = 𝑡 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) |
27 | 21, 23, 26 | 3anbi123d 1432 | . . . 4 ⊢ (𝑤 = 𝑡 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋))) |
28 | wrd2f1tovbij.d | . . . 4 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} | |
29 | 27, 28 | elrab2 3682 | . . 3 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋))) |
30 | preq2 4669 | . . . . 5 ⊢ (𝑛 = (𝑡‘1) → {𝑃, 𝑛} = {𝑃, (𝑡‘1)}) | |
31 | 30 | eleq1d 2897 | . . . 4 ⊢ (𝑛 = (𝑡‘1) → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
32 | wrd2f1tovbij.r | . . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} | |
33 | 31, 32 | elrab2 3682 | . . 3 ⊢ ((𝑡‘1) ∈ 𝑅 ↔ ((𝑡‘1) ∈ 𝑉 ∧ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
34 | 20, 29, 33 | 3imtr4i 294 | . 2 ⊢ (𝑡 ∈ 𝐷 → (𝑡‘1) ∈ 𝑅) |
35 | 1, 34 | fmpti 6875 | 1 ⊢ 𝐹:𝐷⟶𝑅 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 {crab 3142 {cpr 4568 class class class wbr 5065 ↦ cmpt 5145 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 0cc0 10536 1c1 10537 < clt 10674 ℕcn 11637 2c2 11691 ℕ0cn0 11896 ..^cfzo 13032 ♯chash 13689 Word cword 13860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 |
This theorem is referenced by: wwlktovf1 14320 wwlktovfo 14321 |
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