Theorem wlkcompim 26054

Description: Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.)

Hypotheses

Ref	Expression
wlkcomp.1	⊢ 𝐹 = (1st ‘𝑊)
wlkcomp.2	⊢ 𝑃 = (2nd ‘𝑊)

Assertion

Ref	Expression
wlkcompim	⊢ (𝑊 ∈ (𝑉 Walks 𝐸) → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))

Distinct variable groups:   𝑘,𝐸   𝑘,𝐹   𝑃,𝑘   𝑘,𝑉   𝑘,𝑊

Proof of Theorem wlkcompim

Dummy variables 𝑒 𝑓 𝑝 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wlk 26036	. . . 4 ⊢ Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
2		vex 3176	. . . . 5 ⊢ 𝑣 ∈ V
3		vex 3176	. . . . 5 ⊢ 𝑒 ∈ V
4		wlks 26047	. . . . . 6 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → (𝑣 Walks 𝑒) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
5		ovex 6577	. . . . . 6 ⊢ (𝑣 Walks 𝑒) ∈ V
6	4, 5	syl6eqelr 2697	. . . . 5 ⊢ ((𝑣 ∈ V ∧ 𝑒 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
7	2, 3, 6	mp2an 704	. . . 4 ⊢ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} ∈ V
8		dmeq 5246	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
9		wrdeq 13182	. . . . . . . . 9 ⊢ (dom 𝑒 = dom 𝐸 → Word dom 𝑒 = Word dom 𝐸)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → Word dom 𝑒 = Word dom 𝐸)
11	10	eleq2d 2673	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (𝑓 ∈ Word dom 𝑒 ↔ 𝑓 ∈ Word dom 𝐸))
12	11	adantl 481	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑓 ∈ Word dom 𝑒 ↔ 𝑓 ∈ Word dom 𝐸))
13		feq3 5941	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (𝑝:(0...(#‘𝑓))⟶𝑣 ↔ 𝑝:(0...(#‘𝑓))⟶𝑉))
14	13	adantr 480	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑝:(0...(#‘𝑓))⟶𝑣 ↔ 𝑝:(0...(#‘𝑓))⟶𝑉))
15		fveq1 6102	. . . . . . . . 9 ⊢ (𝑒 = 𝐸 → (𝑒‘(𝑓‘𝑘)) = (𝐸‘(𝑓‘𝑘)))
16	15	adantl 481	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑒‘(𝑓‘𝑘)) = (𝐸‘(𝑓‘𝑘)))
17	16	eqeq1d 2612	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
18	17	ralbidv 2969	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
19	12, 14, 18	3anbi123d 1391	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})))
20	19	opabbidv 4648	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})})
21	1, 7, 20	elovmpt2 6777	. . 3 ⊢ (𝑊 ∈ (𝑉 Walks 𝐸) ↔ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}))
22		elopaelxp 5114	. . . 4 ⊢ (𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} → 𝑊 ∈ (V × V))
23		wlkcomp.1	. . . . . 6 ⊢ 𝐹 = (1st ‘𝑊)
24		wlkcomp.2	. . . . . 6 ⊢ 𝑃 = (2nd ‘𝑊)
25	23, 24	wlkcomp 26053	. . . . 5 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ (V × V)) → (𝑊 ∈ (𝑉 Walks 𝐸) ↔ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
26	25	biimpd 218	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ (V × V)) → (𝑊 ∈ (𝑉 Walks 𝐸) → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
27	22, 26	syl3an3 1353	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) → (𝑊 ∈ (𝑉 Walks 𝐸) → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
28	21, 27	sylbi 206	. 2 ⊢ (𝑊 ∈ (𝑉 Walks 𝐸) → (𝑊 ∈ (𝑉 Walks 𝐸) → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
29	28	pm2.43i 50	1 ⊢ (𝑊 ∈ (𝑉 Walks 𝐸) → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  {cpr 4127  {copab 4642   × cxp 5036  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   Walks cwalk 26026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036

This theorem is referenced by:  usg2wlkeq  26236