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| Mirrors > Home > MPE Home > Th. List > wwlktovf | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for wrd2f1tovbij 14999. (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 14557 | . . . . 5 ⊢ (𝑡 ∈ Word 𝑉 → 𝑡:(0..^(♯‘𝑡))⟶𝑉) | |
| 3 | oveq2 7439 | . . . . . . . 8 ⊢ ((♯‘𝑡) = 2 → (0..^(♯‘𝑡)) = (0..^2)) | |
| 4 | 3 | feq2d 6722 | . . . . . . 7 ⊢ ((♯‘𝑡) = 2 → (𝑡:(0..^(♯‘𝑡))⟶𝑉 ↔ 𝑡:(0..^2)⟶𝑉)) |
| 5 | 1nn0 12542 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
| 6 | 2nn 12339 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
| 7 | 1lt2 12437 | . . . . . . . . 9 ⊢ 1 < 2 | |
| 8 | elfzo0 13740 | . . . . . . . . 9 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2)) | |
| 9 | 5, 6, 7, 8 | mpbir3an 1342 | . . . . . . . 8 ⊢ 1 ∈ (0..^2) |
| 10 | ffvelcdm 7101 | . . . . . . . 8 ⊢ ((𝑡:(0..^2)⟶𝑉 ∧ 1 ∈ (0..^2)) → (𝑡‘1) ∈ 𝑉) | |
| 11 | 9, 10 | mpan2 691 | . . . . . . 7 ⊢ (𝑡:(0..^2)⟶𝑉 → (𝑡‘1) ∈ 𝑉) |
| 12 | 4, 11 | biimtrdi 253 | . . . . . 6 ⊢ ((♯‘𝑡) = 2 → (𝑡:(0..^(♯‘𝑡))⟶𝑉 → (𝑡‘1) ∈ 𝑉)) |
| 13 | 12 | 3ad2ant1 1134 | . . . . 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 4733 | . . . . . . . 8 ⊢ ((𝑡‘0) = 𝑃 → {(𝑡‘0), (𝑡‘1)} = {𝑃, (𝑡‘1)}) | |
| 16 | 15 | eleq1d 2826 | . . . . . . 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 1131 | . . . . 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 6915 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((♯‘𝑤) = 2 ↔ (♯‘𝑡) = 2)) | |
| 22 | fveq1 6905 | . . . . . 6 ⊢ (𝑤 = 𝑡 → (𝑤‘0) = (𝑡‘0)) | |
| 23 | 22 | eqeq1d 2739 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((𝑤‘0) = 𝑃 ↔ (𝑡‘0) = 𝑃)) |
| 24 | fveq1 6905 | . . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤‘1) = (𝑡‘1)) | |
| 25 | 22, 24 | preq12d 4741 | . . . . . 6 ⊢ (𝑤 = 𝑡 → {(𝑤‘0), (𝑤‘1)} = {(𝑡‘0), (𝑡‘1)}) |
| 26 | 25 | eleq1d 2826 | . . . . 5 ⊢ (𝑤 = 𝑡 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) |
| 27 | 21, 23, 26 | 3anbi123d 1438 | . . . 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 3695 | . . 3 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋))) |
| 30 | preq2 4734 | . . . . 5 ⊢ (𝑛 = (𝑡‘1) → {𝑃, 𝑛} = {𝑃, (𝑡‘1)}) | |
| 31 | 30 | eleq1d 2826 | . . . 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 3695 | . . 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 7132 | 1 ⊢ 𝐹:𝐷⟶𝑅 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 < clt 11295 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ..^cfzo 13694 ♯chash 14369 Word cword 14552 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 |
| This theorem is referenced by: wwlktovf1 14996 wwlktovfo 14997 |
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