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Theorem wlkcomp 27672

Description: A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)

**Hypotheses**
Ref	Expression
wlkcomp.v	⊢ 𝑉 = (Vtx‘𝐺)
wlkcomp.i	⊢ 𝐼 = (iEdg‘𝐺)
wlkcomp.1	⊢ 𝐹 = (1^st ‘𝑊)
wlkcomp.2	⊢ 𝑃 = (2^nd ‘𝑊)

**Assertion**
Ref	Expression
wlkcomp	⊢ ((𝐺 ∈ 𝑈 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (Walks‘𝐺) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))

Distinct variable groups: 𝑘,𝐹 𝑘,𝐺 𝑃,𝑘 Allowed substitution hints: 𝑆(𝑘) 𝑇(𝑘) 𝑈(𝑘) 𝐼(𝑘) 𝑉(𝑘) 𝑊(𝑘)

**Proof of Theorem wlkcomp**
Step	Hyp	Ref	Expression
1		wlkcomp.1	. . . . . . 7 ⊢ 𝐹 = (1^st ‘𝑊)
2	1	eqcomi 2745	. . . . . 6 ⊢ (1^st ‘𝑊) = 𝐹
3		wlkcomp.2	. . . . . . 7 ⊢ 𝑃 = (2^nd ‘𝑊)
4	3	eqcomi 2745	. . . . . 6 ⊢ (2^nd ‘𝑊) = 𝑃
5	2, 4	pm3.2i 474	. . . . 5 ⊢ ((1^st ‘𝑊) = 𝐹 ∧ (2^nd ‘𝑊) = 𝑃)
6		eqop 7781	. . . . 5 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 = ⟨𝐹, 𝑃⟩ ↔ ((1^st ‘𝑊) = 𝐹 ∧ (2^nd ‘𝑊) = 𝑃)))
7	5, 6	mpbiri 261	. . . 4 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → 𝑊 = ⟨𝐹, 𝑃⟩)
8	7	eleq1d 2815	. . 3 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺)))
9		df-br 5040	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺))
10	8, 9	bitr4di 292	. 2 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (Walks‘𝐺) ↔ 𝐹(Walks‘𝐺)𝑃))
11		wlkcomp.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
12		wlkcomp.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
13	11, 12	iswlkg 27655	. 2 ⊢ (𝐺 ∈ 𝑈 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
14	10, 13	sylan9bbr 514	1 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (Walks‘𝐺) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 if-wif 1063 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ⊆ wss 3853 {csn 4527 {cpr 4529 ⟨cop 4533 class class class wbr 5039 × cxp 5534 dom cdm 5536 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 1^st c1st 7737 2^nd c2nd 7738 0cc0 10694 1c1 10695 + caddc 10697 ...cfz 13060 ..^cfzo 13203 ♯chash 13861 Word cword 14034 Vtxcvtx 27041 iEdgciedg 27042 Walkscwlks 27638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-wlks 27641

This theorem is referenced by: wlkcompim 27673

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