Theorem wlkelwrd 26058

Description: The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.)

Assertion

Ref	Expression
wlkelwrd	⊢ (𝑊 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉))

Proof of Theorem wlkelwrd

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

Step	Hyp	Ref	Expression
1		df-wlk 26036	. . 3 ⊢ 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	. . . 4 ⊢ 𝑣 ∈ V
3		vex 3176	. . . 4 ⊢ 𝑒 ∈ V
4		wlks 26047	. . . . 5 ⊢ ((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	. . . . 5 ⊢ (𝑣 Walks 𝑒) ∈ V
6	4, 5	syl6eqelr 2697	. . . 4 ⊢ ((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	. . 3 ⊢ {⟨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	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
9		wrdeq 13182	. . . . . . . 8 ⊢ (dom 𝑒 = dom 𝐸 → Word dom 𝑒 = Word dom 𝐸)
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝑒 = 𝐸 → Word dom 𝑒 = Word dom 𝐸)
11	10	eleq2d 2673	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝑓 ∈ Word dom 𝑒 ↔ 𝑓 ∈ Word dom 𝐸))
12	11	adantl 481	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑓 ∈ Word dom 𝑒 ↔ 𝑓 ∈ Word dom 𝐸))
13		feq3 5941	. . . . . 6 ⊢ (𝑣 = 𝑉 → (𝑝:(0...(#‘𝑓))⟶𝑣 ↔ 𝑝:(0...(#‘𝑓))⟶𝑉))
14	13	adantr 480	. . . . 5 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑝:(0...(#‘𝑓))⟶𝑣 ↔ 𝑝:(0...(#‘𝑓))⟶𝑉))
15		fveq1 6102	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (𝑒‘(𝑓‘𝑘)) = (𝐸‘(𝑓‘𝑘)))
16	15	adantl 481	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑒‘(𝑓‘𝑘)) = (𝐸‘(𝑓‘𝑘)))
17	16	eqeq1d 2612	. . . . . 6 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
18	17	ralbidv 2969	. . . . 5 ⊢ ((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	. . . 4 ⊢ ((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	. . 3 ⊢ ((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	. 2 ⊢ (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		eleq1 2676	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑊) → (𝑓 ∈ Word dom 𝐸 ↔ (1st ‘𝑊) ∈ Word dom 𝐸))
23		fveq2 6103	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑊) → (#‘𝑓) = (#‘(1st ‘𝑊)))
24	23	oveq2d 6565	. . . . . . 7 ⊢ (𝑓 = (1st ‘𝑊) → (0...(#‘𝑓)) = (0...(#‘(1st ‘𝑊))))
25	24	feq2d 5944	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑊) → (𝑝:(0...(#‘𝑓))⟶𝑉 ↔ 𝑝:(0...(#‘(1st ‘𝑊)))⟶𝑉))
26	23	oveq2d 6565	. . . . . . 7 ⊢ (𝑓 = (1st ‘𝑊) → (0..^(#‘𝑓)) = (0..^(#‘(1st ‘𝑊))))
27		fveq1 6102	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑊) → (𝑓‘𝑘) = ((1st ‘𝑊)‘𝑘))
28	27	fveq2d 6107	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑊) → (𝐸‘(𝑓‘𝑘)) = (𝐸‘((1st ‘𝑊)‘𝑘)))
29	28	eqeq1d 2612	. . . . . . 7 ⊢ (𝑓 = (1st ‘𝑊) → ((𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘((1st ‘𝑊)‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
30	26, 29	raleqbidv 3129	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑊) → (∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}))
31	22, 25, 30	3anbi123d 1391	. . . . 5 ⊢ (𝑓 = (1st ‘𝑊) → ((𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) ↔ ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘(1st ‘𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})))
32		feq1 5939	. . . . . 6 ⊢ (𝑝 = (2nd ‘𝑊) → (𝑝:(0...(#‘(1st ‘𝑊)))⟶𝑉 ↔ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉))
33		fveq1 6102	. . . . . . . . 9 ⊢ (𝑝 = (2nd ‘𝑊) → (𝑝‘𝑘) = ((2nd ‘𝑊)‘𝑘))
34		fveq1 6102	. . . . . . . . 9 ⊢ (𝑝 = (2nd ‘𝑊) → (𝑝‘(𝑘 + 1)) = ((2nd ‘𝑊)‘(𝑘 + 1)))
35	33, 34	preq12d 4220	. . . . . . . 8 ⊢ (𝑝 = (2nd ‘𝑊) → {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} = {((2nd ‘𝑊)‘𝑘), ((2nd ‘𝑊)‘(𝑘 + 1))})
36	35	eqeq2d 2620	. . . . . . 7 ⊢ (𝑝 = (2nd ‘𝑊) → ((𝐸‘((1st ‘𝑊)‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘((1st ‘𝑊)‘𝑘)) = {((2nd ‘𝑊)‘𝑘), ((2nd ‘𝑊)‘(𝑘 + 1))}))
37	36	ralbidv 2969	. . . . . 6 ⊢ (𝑝 = (2nd ‘𝑊) → (∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {((2nd ‘𝑊)‘𝑘), ((2nd ‘𝑊)‘(𝑘 + 1))}))
38	32, 37	3anbi23d 1394	. . . . 5 ⊢ (𝑝 = (2nd ‘𝑊) → (((1st ‘𝑊) ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘(1st ‘𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) ↔ ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {((2nd ‘𝑊)‘𝑘), ((2nd ‘𝑊)‘(𝑘 + 1))})))
39	31, 38	elopabi 7120	. . . 4 ⊢ (𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} → ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {((2nd ‘𝑊)‘𝑘), ((2nd ‘𝑊)‘(𝑘 + 1))}))
40		3simpa 1051	. . . 4 ⊢ (((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st ‘𝑊)))(𝐸‘((1st ‘𝑊)‘𝑘)) = {((2nd ‘𝑊)‘𝑘), ((2nd ‘𝑊)‘(𝑘 + 1))}) → ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉))
41	39, 40	syl 17	. . 3 ⊢ (𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} → ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉))
42	41	3ad2ant3 1077	. 2 ⊢ ((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))})}) → ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉))
43	21, 42	sylbi 206	1 ⊢ (𝑊 ∈ (𝑉 Walks 𝐸) → ((1st ‘𝑊) ∈ Word dom 𝐸 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶𝑉))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  {cpr 4127  {copab 4642  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-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:  vfwlkniswwlkn  26234  2wlkeq  26235  usg2wlkeq  26236