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Mirrors > Home > MPE Home > Th. List > wwlktovf | Structured version Visualization version GIF version |
Description: Lemma 1 for wrd2f1tovbij 13551. (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 13165 | . . . . 5 ⊢ (𝑡 ∈ Word 𝑉 → 𝑡:(0..^(#‘𝑡))⟶𝑉) | |
3 | oveq2 6557 | . . . . . . . 8 ⊢ ((#‘𝑡) = 2 → (0..^(#‘𝑡)) = (0..^2)) | |
4 | 3 | feq2d 5944 | . . . . . . 7 ⊢ ((#‘𝑡) = 2 → (𝑡:(0..^(#‘𝑡))⟶𝑉 ↔ 𝑡:(0..^2)⟶𝑉)) |
5 | 1nn0 11185 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
6 | 2nn 11062 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
7 | 1lt2 11071 | . . . . . . . . 9 ⊢ 1 < 2 | |
8 | elfzo0 12376 | . . . . . . . . 9 ⊢ (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2)) | |
9 | 5, 6, 7, 8 | mpbir3an 1237 | . . . . . . . 8 ⊢ 1 ∈ (0..^2) |
10 | ffvelrn 6265 | . . . . . . . 8 ⊢ ((𝑡:(0..^2)⟶𝑉 ∧ 1 ∈ (0..^2)) → (𝑡‘1) ∈ 𝑉) | |
11 | 9, 10 | mpan2 703 | . . . . . . 7 ⊢ (𝑡:(0..^2)⟶𝑉 → (𝑡‘1) ∈ 𝑉) |
12 | 4, 11 | syl6bi 242 | . . . . . 6 ⊢ ((#‘𝑡) = 2 → (𝑡:(0..^(#‘𝑡))⟶𝑉 → (𝑡‘1) ∈ 𝑉)) |
13 | 12 | 3ad2ant1 1075 | . . . . 5 ⊢ (((#‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → (𝑡:(0..^(#‘𝑡))⟶𝑉 → (𝑡‘1) ∈ 𝑉)) |
14 | 2, 13 | mpan9 485 | . . . 4 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → (𝑡‘1) ∈ 𝑉) |
15 | preq1 4212 | . . . . . . . 8 ⊢ ((𝑡‘0) = 𝑃 → {(𝑡‘0), (𝑡‘1)} = {𝑃, (𝑡‘1)}) | |
16 | 15 | eleq1d 2672 | . . . . . . 7 ⊢ ((𝑡‘0) = 𝑃 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
17 | 16 | biimpa 500 | . . . . . 6 ⊢ (((𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) → {𝑃, (𝑡‘1)} ∈ 𝑋) |
18 | 17 | 3adant1 1072 | . . . . 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 553 | . . 3 ⊢ ((𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) → ((𝑡‘1) ∈ 𝑉 ∧ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
21 | fveq2 6103 | . . . . . 6 ⊢ (𝑤 = 𝑡 → (#‘𝑤) = (#‘𝑡)) | |
22 | 21 | eqeq1d 2612 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((#‘𝑤) = 2 ↔ (#‘𝑡) = 2)) |
23 | fveq1 6102 | . . . . . 6 ⊢ (𝑤 = 𝑡 → (𝑤‘0) = (𝑡‘0)) | |
24 | 23 | eqeq1d 2612 | . . . . 5 ⊢ (𝑤 = 𝑡 → ((𝑤‘0) = 𝑃 ↔ (𝑡‘0) = 𝑃)) |
25 | fveq1 6102 | . . . . . . 7 ⊢ (𝑤 = 𝑡 → (𝑤‘1) = (𝑡‘1)) | |
26 | 23, 25 | preq12d 4220 | . . . . . 6 ⊢ (𝑤 = 𝑡 → {(𝑤‘0), (𝑤‘1)} = {(𝑡‘0), (𝑡‘1)}) |
27 | 26 | eleq1d 2672 | . . . . 5 ⊢ (𝑤 = 𝑡 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)) |
28 | 22, 24, 27 | 3anbi123d 1391 | . . . 4 ⊢ (𝑤 = 𝑡 → (((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((#‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋))) |
29 | wrd2f1tovbij.d | . . . 4 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} | |
30 | 28, 29 | elrab2 3333 | . . 3 ⊢ (𝑡 ∈ 𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋))) |
31 | preq2 4213 | . . . . 5 ⊢ (𝑛 = (𝑡‘1) → {𝑃, 𝑛} = {𝑃, (𝑡‘1)}) | |
32 | 31 | eleq1d 2672 | . . . 4 ⊢ (𝑛 = (𝑡‘1) → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
33 | wrd2f1tovbij.r | . . . 4 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} | |
34 | 32, 33 | elrab2 3333 | . . 3 ⊢ ((𝑡‘1) ∈ 𝑅 ↔ ((𝑡‘1) ∈ 𝑉 ∧ {𝑃, (𝑡‘1)} ∈ 𝑋)) |
35 | 20, 30, 34 | 3imtr4i 280 | . 2 ⊢ (𝑡 ∈ 𝐷 → (𝑡‘1) ∈ 𝑅) |
36 | 1, 35 | fmpti 6291 | 1 ⊢ 𝐹:𝐷⟶𝑅 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 < clt 9953 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ..^cfzo 12334 #chash 12979 Word cword 13146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 |
This theorem is referenced by: wwlktovf1 13548 wwlktovfo 13549 |
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