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Theorem wwlksnextbij0 28167

Description: Lemma for wwlksnextbij 28168. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
wwlksnextbij0	⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅)

Distinct variable groups: 𝑤,𝐺 𝑤,𝑁 𝑤,𝑊 𝑡,𝐷 𝑛,𝐸,𝑤 𝑡,𝑁,𝑤 𝑡,𝑅 𝑛,𝑉,𝑤 𝑛,𝑊 𝑡,𝑛,𝑁,𝑤 Allowed substitution hints: 𝐷(𝑤,𝑛) 𝑅(𝑤,𝑛) 𝐸(𝑡) 𝐹(𝑤,𝑡,𝑛) 𝐺(𝑡,𝑛) 𝑉(𝑡) 𝑊(𝑡)

**Proof of Theorem wwlksnextbij0**
Step	Hyp	Ref	Expression
1		wwlksnextbij0.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2	1	wwlknbp 28108	. . 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 28165	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷–1-1→𝑅)
8	7	3ad2ant2 1132	. . 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 28166	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)
11		df-f1o 6425	. 2 ⊢ (𝐹:𝐷–1-1-onto→𝑅 ↔ (𝐹:𝐷–1-1→𝑅 ∧ 𝐹:𝐷–onto→𝑅))
12	9, 10, 11	sylanbrc 582	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 {crab 3067 Vcvv 3422 {cpr 4560 ↦ cmpt 5153 –1-1→wf1 6415 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 1c1 10803 + caddc 10805 2c2 11958 ℕ₀cn0 12163 ♯chash 13972 Word cword 14145 lastSclsw 14193 prefix cpfx 14311 Vtxcvtx 27269 Edgcedg 27320 WWalksN cwwlksn 28092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312 df-wwlks 28096 df-wwlksn 28097

This theorem is referenced by: wwlksnextbij 28168

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