Theorem numclwwlkovfel2 26610

Description: Properties of an element of the value of operation 𝐹. (Contributed by Alexander van der Vekens, 20-Sep-2018.)

Hypotheses

Ref	Expression
numclwwlk.c	⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
numclwwlk.f	⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})

Assertion

Ref	Expression
numclwwlkovfel2	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝑋𝐹𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋)))

Distinct variable groups:   𝑛,𝐸   𝑛,𝑁   𝑛,𝑉   𝑖,𝐸   𝑖,𝑁   𝑖,𝑉   𝑤,𝑖   𝑤,𝐶   𝑤,𝑁   𝐶,𝑛,𝑣,𝑤   𝑣,𝑁   𝑛,𝑋,𝑣,𝑤   𝑣,𝑉   𝐴,𝑖,𝑤

Allowed substitution hints:   𝐴(𝑣,𝑛)   𝐶(𝑖)   𝐸(𝑤,𝑣)   𝐹(𝑤,𝑣,𝑖,𝑛)   𝑉(𝑤)   𝑋(𝑖)

Proof of Theorem numclwwlkovfel2

Step	Hyp	Ref	Expression
1		pm3.22 464	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0))
2	1	3adant1 1072	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0))
3		numclwwlk.c	. . . . 5 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))
4		numclwwlk.f	. . . . 5 ⊢ 𝐹 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶‘𝑛) ∣ (𝑤‘0) = 𝑣})
5	3, 4	numclwwlkovf 26608	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑋})
6	2, 5	syl 17	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑋})
7	6	eleq2d 2673	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝑋𝐹𝑁) ↔ 𝐴 ∈ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑋}))
8	3	numclwwlkfvc 26604	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝐶‘𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁))
9	8	3ad2ant2 1076	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐶‘𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁))
10	9	eleq2d 2673	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝐶‘𝑁) ↔ 𝐴 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
11		usgrav 25867	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
12	11	anim1i 590	. . . . . . . 8 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
13		df-3an 1033	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
14	12, 13	sylibr 223	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
15	14	3adant3 1074	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
16		isclwwlkn 26297	. . . . . 6 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐴 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝐴) = 𝑁)))
17	15, 16	syl 17	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝐴 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝐴) = 𝑁)))
18		isclwwlk 26296	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝐴 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸)))
19	11, 18	syl 17	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (𝐴 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸)))
20	19	3ad2ant1 1075	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸)))
21	20	anbi1d 737	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → ((𝐴 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝐴) = 𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁)))
22	10, 17, 21	3bitrd 293	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝐶‘𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁)))
23	22	anbi1d 737	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → ((𝐴 ∈ (𝐶‘𝑁) ∧ (𝐴‘0) = 𝑋) ↔ (((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁) ∧ (𝐴‘0) = 𝑋)))
24		fveq1 6102	. . . . 5 ⊢ (𝑤 = 𝐴 → (𝑤‘0) = (𝐴‘0))
25	24	eqeq1d 2612	. . . 4 ⊢ (𝑤 = 𝐴 → ((𝑤‘0) = 𝑋 ↔ (𝐴‘0) = 𝑋))
26	25	elrab 3331	. . 3 ⊢ (𝐴 ∈ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑋} ↔ (𝐴 ∈ (𝐶‘𝑁) ∧ (𝐴‘0) = 𝑋))
27		df-3an 1033	. . 3 ⊢ (((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋) ↔ (((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁) ∧ (𝐴‘0) = 𝑋))
28	23, 26, 27	3bitr4g 302	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ {𝑤 ∈ (𝐶‘𝑁) ∣ (𝑤‘0) = 𝑋} ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋)))
29	7, 28	bitrd 267	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉) → (𝐴 ∈ (𝑋𝐹𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ₀cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   USGrph cusg 25859   ClWWalks cclwwlk 26276   ClWWalksN cclwwlkn 26277

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-usgra 25862  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  numclwwlkovf2ex  26613  numclwlk1lem2foa  26618  numclwlk1lem2fo  26622