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Mirrors > Home > MPE Home > Th. List > wwlktovf | Structured version Visualization version GIF version |
Description: Lemma 1 for wrd2f1tovbij 14917. (Contributed by Alexander van der Vekens, 27-Jul-2018.) |
Ref | Expression |
---|---|
wwlktovf1o.d | ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} |
wwlktovf1o.r | ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} |
wwlktovf1o.f | ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) |
Ref | Expression |
---|---|
wwlktovf | ⊢ 𝐹:𝐷⟶𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlktovf1o.f | . 2 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡‘1)) | |
2 | wrdf 14475 | . . . . 5 ⊢ (𝑡 ∈ Word 𝑉 → 𝑡:(0..^(♯‘𝑡))⟶𝑉) | |
3 | oveq2 7413 | . . . . . . . 8 ⊢ ((♯‘𝑡) = 2 → (0..^(♯‘𝑡)) = (0..^2)) | |
4 | 3 | feq2d 6697 | . . . . . . 7 ⊢ ((♯‘𝑡) = 2 → (𝑡:(0..^(♯‘𝑡))⟶𝑉 ↔ 𝑡:(0..^2)⟶𝑉)) |
5 | 1nn0 12492 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
6 | 2nn 12289 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
7 | 1lt2 12387 | . . . . . . . . 9 ⊢ 1 < 2 | |
8 | elfzo0 13679 | . . . . . . . . 9 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2)) | |
9 | 5, 6, 7, 8 | mpbir3an 1338 | . . . . . . . 8 ⊢ 1 ∈ (0..^2) |
10 | ffvelcdm 7077 | . . . . . . . 8 ⊢ ((𝑡:(0..^2)⟶𝑉 ∧ 1 ∈ (0..^2)) → (𝑡‘1) ∈ 𝑉) | |
11 | 9, 10 | mpan2 688 | . . . . . . 7 ⊢ (𝑡:(0..^2)⟶𝑉 → (𝑡‘1) ∈ 𝑉) |
12 | 4, 11 | biimtrdi 252 | . . . . . 6 ⊢ ((♯‘𝑡) = 2 → (𝑡:(0..^(♯‘𝑡))⟶𝑉 → (𝑡‘1) ∈ 𝑉)) |
13 | 12 | 3ad2ant1 1130 | . . . . 5 ⊢ (((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → (𝑡:(0..^(♯‘𝑡))⟶𝑉 → (𝑡‘1) ∈ 𝑉)) |
14 | 2, 13 | mpan9 506 | . . . 4 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → (𝑡‘1) ∈ 𝑉) |
15 | preq1 4732 | . . . . . . . 8 ⊢ ((𝑡‘0) = 𝑃 → {(𝑡‘0), (𝑡‘1)} = {𝑃, (𝑡‘1)}) | |
16 | 15 | eleq1d 2812 | . . . . . . 7 ⊢ ((𝑡‘0) = 𝑃 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
17 | 16 | biimpa 476 | . . . . . 6 ⊢ (((𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → {𝑃, (𝑡‘1)} ∈ 𝑋) |
18 | 17 | 3adant1 1127 | . . . . 5 ⊢ (((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → {𝑃, (𝑡‘1)} ∈ 𝑋) |
19 | 18 | adantl 481 | . . . 4 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → {𝑃, (𝑡‘1)} ∈ 𝑋) |
20 | 14, 19 | jca 511 | . . 3 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → ((𝑡‘1) ∈ 𝑉 ∧ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
21 | fveqeq2 6894 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((♯‘𝑤) = 2 ↔ (♯‘𝑡) = 2)) | |
22 | fveq1 6884 | . . . . . 6 ⊢ (𝑤 = 𝑡 → (𝑤‘0) = (𝑡‘0)) | |
23 | 22 | eqeq1d 2728 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((𝑤‘0) = 𝑃 ↔ (𝑡‘0) = 𝑃)) |
24 | fveq1 6884 | . . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤‘1) = (𝑡‘1)) | |
25 | 22, 24 | preq12d 4740 | . . . . . 6 ⊢ (𝑤 = 𝑡 → {(𝑤‘0), (𝑤‘1)} = {(𝑡‘0), (𝑡‘1)}) |
26 | 25 | eleq1d 2812 | . . . . 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 | wwlktovf1o.d | . . . 4 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} | |
29 | 27, 28 | elrab2 3681 | . . 3 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋))) |
30 | preq2 4733 | . . . . 5 ⊢ (𝑛 = (𝑡‘1) → {𝑃, 𝑛} = {𝑃, (𝑡‘1)}) | |
31 | 30 | eleq1d 2812 | . . . 4 ⊢ (𝑛 = (𝑡‘1) → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
32 | wwlktovf1o.r | . . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} | |
33 | 31, 32 | elrab2 3681 | . . 3 ⊢ ((𝑡‘1) ∈ 𝑅 ↔ ((𝑡‘1) ∈ 𝑉 ∧ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
34 | 20, 29, 33 | 3imtr4i 292 | . 2 ⊢ (𝑡 ∈ 𝐷 → (𝑡‘1) ∈ 𝑅) |
35 | 1, 34 | fmpti 7107 | 1 ⊢ 𝐹:𝐷⟶𝑅 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3426 {cpr 4625 class class class wbr 5141 ↦ cmpt 5224 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 < clt 11252 ℕcn 12216 2c2 12271 ℕ0cn0 12476 ..^cfzo 13633 ♯chash 14295 Word cword 14470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 |
This theorem is referenced by: wwlktovf1 14914 wwlktovfo 14915 |
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