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Theorem wwlksnextbij0 30101

Description: Lemma for wwlksnextbij 30102. (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 30042	. . 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 30099	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 𝐹:𝐷–1-1→𝑅)
8	7	3ad2ant2 1147	. . 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 30100	. 2 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)
11		df-f1o 6528	. 2 ⊢ (𝐹:𝐷–1-1-onto→𝑅 ↔ (𝐹:𝐷–1-1→𝑅 ∧ 𝐹:𝐷–onto→𝑅))
12	9, 10, 11	sylanbrc 592	1 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 {crab 3414 Vcvv 3454 {cpr 4584 ↦ cmpt 5181 –1-1→wf1 6518 –onto→wfo 6519 –1-1-onto→wf1o 6520 ‘cfv 6521 (class class class)co 7396 1c1 11074 + caddc 11076 2c2 12272 ℕ₀cn0 12481 ♯chash 14343 Word cword 14526 lastSclsw 14575 prefix cpfx 14684 Vtxcvtx 29197 Edgcedg 29248 WWalksN cwwlksn 30026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-2 12280 df-n0 12482 df-xnn0 12555 df-z 12569 df-uz 12840 df-rp 12994 df-fz 13513 df-fzo 13660 df-hash 14344 df-word 14527 df-lsw 14576 df-concat 14584 df-s1 14610 df-substr 14655 df-pfx 14685 df-wwlks 30030 df-wwlksn 30031

This theorem is referenced by: wwlksnextbij 30102

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