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Mirrors > Home > MPE Home > Th. List > wwlksnextbij0 | Structured version Visualization version GIF version |
Description: Lemma for wwlksnextbij 29135. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) |
Ref | Expression |
---|---|
wwlksnextbij0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wwlksnextbij0.e | ⊢ 𝐸 = (Edg‘𝐺) |
wwlksnextbij0.d | ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} |
wwlksnextbij0.r | ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} |
wwlksnextbij0.f | ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡)) |
Ref | Expression |
---|---|
wwlksnextbij0 | ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlksnextbij0.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | wwlknbp 29075 | . . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉)) |
3 | wwlksnextbij0.e | . . . . 5 ⊢ 𝐸 = (Edg‘𝐺) | |
4 | wwlksnextbij0.d | . . . . 5 ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)} | |
5 | wwlksnextbij0.r | . . . . 5 ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸} | |
6 | wwlksnextbij0.f | . . . . 5 ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡)) | |
7 | 1, 3, 4, 5, 6 | wwlksnextinj 29132 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷–1-1→𝑅) |
8 | 7 | 3ad2ant2 1135 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉) → 𝐹:𝐷–1-1→𝑅) |
9 | 2, 8 | syl 17 | . 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1→𝑅) |
10 | 1, 3, 4, 5, 6 | wwlksnextsurj 29133 | . 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅) |
11 | df-f1o 6546 | . 2 ⊢ (𝐹:𝐷–1-1-onto→𝑅 ↔ (𝐹:𝐷–1-1→𝑅 ∧ 𝐹:𝐷–onto→𝑅)) | |
12 | 9, 10, 11 | sylanbrc 584 | 1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 {cpr 4628 ↦ cmpt 5229 –1-1→wf1 6536 –onto→wfo 6537 –1-1-onto→wf1o 6538 ‘cfv 6539 (class class class)co 7403 1c1 11106 + caddc 11108 2c2 12262 ℕ0cn0 12467 ♯chash 14285 Word cword 14459 lastSclsw 14507 prefix cpfx 14615 Vtxcvtx 28235 Edgcedg 28286 WWalksN cwwlksn 29059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-n0 12468 df-xnn0 12540 df-z 12554 df-uz 12818 df-rp 12970 df-fz 13480 df-fzo 13623 df-hash 14286 df-word 14460 df-lsw 14508 df-concat 14516 df-s1 14541 df-substr 14586 df-pfx 14616 df-wwlks 29063 df-wwlksn 29064 |
This theorem is referenced by: wwlksnextbij 29135 |
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