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Theorem wlkcomp 29699

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 470	. . . . 5 ⊢ ((1^st ‘𝑊) = 𝐹 ∧ (2^nd ‘𝑊) = 𝑃)
6		eqop 7984	. . . . 5 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 = ⟨𝐹, 𝑃⟩ ↔ ((1^st ‘𝑊) = 𝐹 ∧ (2^nd ‘𝑊) = 𝑃)))
7	5, 6	mpbiri 258	. . . 4 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → 𝑊 = ⟨𝐹, 𝑃⟩)
8	7	eleq1d 2821	. . 3 ⊢ (𝑊 ∈ (𝑆 × 𝑇) → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺)))
9		df-br 5086	. . 3 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺))
10	8, 9	bitr4di 289	. 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 29682	. 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 510	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 206 ∧ wa 395 if-wif 1063 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3051 ⊆ wss 3889 {csn 4567 {cpr 4569 ⟨cop 4573 class class class wbr 5085 × cxp 5629 dom cdm 5631 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 0cc0 11038 1c1 11039 + caddc 11041 ...cfz 13461 ..^cfzo 13608 ♯chash 14292 Word cword 14475 Vtxcvtx 29065 iEdgciedg 29066 Walkscwlks 29665

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-pm 8776 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476 df-wlks 29668

This theorem is referenced by: wlkcompim 29700

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