Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > wwlksnextbij0 | Structured version Visualization version GIF version |
Description: Lemma for wwlksnextbij 27678. (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 27618 | . . 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 27675 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷–1-1→𝑅) |
8 | 7 | 3ad2ant2 1129 | . . 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 27676 | . 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅) |
11 | df-f1o 6355 | . 2 ⊢ (𝐹:𝐷–1-1-onto→𝑅 ↔ (𝐹:𝐷–1-1→𝑅 ∧ 𝐹:𝐷–onto→𝑅)) | |
12 | 9, 10, 11 | sylanbrc 585 | 1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 {crab 3141 Vcvv 3491 {cpr 4562 ↦ cmpt 5139 –1-1→wf1 6345 –onto→wfo 6346 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7149 1c1 10531 + caddc 10533 2c2 11686 ℕ0cn0 11891 ♯chash 13687 Word cword 13858 lastSclsw 13909 prefix cpfx 14027 Vtxcvtx 26779 Edgcedg 26830 WWalksN cwwlksn 27602 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-hash 13688 df-word 13859 df-lsw 13910 df-concat 13918 df-s1 13945 df-substr 13998 df-pfx 14028 df-wwlks 27606 df-wwlksn 27607 |
This theorem is referenced by: wwlksnextbij 27678 |
Copyright terms: Public domain | W3C validator |