Theorem is2wlk 26095

Description: Properties of a pair of functions to be a walk of length 2 (in an undirected graph). (Contributed by Alexander van der Vekens, 16-Feb-2018.)

Assertion

Ref	Expression
is2wlk	⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Proof of Theorem is2wlk

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iswlk 26048	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})))
2	1	anbi1d 737	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (#‘𝐹) = 2)))
3		wrdf 13165	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word dom 𝐸 → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
4		oveq2 6557	. . . . . . . . . . 11 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
5	4	feq2d 5944	. . . . . . . . . 10 ⊢ ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸 ↔ 𝐹:(0..^2)⟶dom 𝐸))
6	3, 5	syl5ib 233	. . . . . . . . 9 ⊢ ((#‘𝐹) = 2 → (𝐹 ∈ Word dom 𝐸 → 𝐹:(0..^2)⟶dom 𝐸))
7		iswrdi 13164	. . . . . . . . 9 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → 𝐹 ∈ Word dom 𝐸)
8	6, 7	impbid1 214	. . . . . . . 8 ⊢ ((#‘𝐹) = 2 → (𝐹 ∈ Word dom 𝐸 ↔ 𝐹:(0..^2)⟶dom 𝐸))
9	8	adantl 481	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ (#‘𝐹) = 2) → (𝐹 ∈ Word dom 𝐸 ↔ 𝐹:(0..^2)⟶dom 𝐸))
10		oveq2 6557	. . . . . . . . 9 ⊢ ((#‘𝐹) = 2 → (0...(#‘𝐹)) = (0...2))
11	10	feq2d 5944	. . . . . . . 8 ⊢ ((#‘𝐹) = 2 → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
12	11	adantl 481	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ (#‘𝐹) = 2) → (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉))
13		fzo0to2pr 12420	. . . . . . . . . . 11 ⊢ (0..^2) = {0, 1}
14	4, 13	syl6eq 2660	. . . . . . . . . 10 ⊢ ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
15	14	raleqdv 3121	. . . . . . . . 9 ⊢ ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))
16		2wlklem 26094	. . . . . . . . 9 ⊢ (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
17	15, 16	syl6bb 275	. . . . . . . 8 ⊢ ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
18	17	adantl 481	. . . . . . 7 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ (#‘𝐹) = 2) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
19	9, 12, 18	3anbi123d 1391	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ (#‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
20		3anass 1035	. . . . . 6 ⊢ ((𝐹:(0..^2)⟶dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
21	19, 20	syl6bb 275	. . . . 5 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) ∧ (#‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))))
22	21	ex 449	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → ((#‘𝐹) = 2 → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))))
23	22	pm5.32rd 670	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (#‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐸 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (#‘𝐹) = 2)))
24		3anass 1035	. . . 4 ⊢ (((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
25		an32 835	. . . 4 ⊢ (((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ↔ ((𝐹:(0..^2)⟶dom 𝐸 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (#‘𝐹) = 2))
26	24, 25	bitri 263	. . 3 ⊢ (((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐸 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (#‘𝐹) = 2))
27	23, 26	syl6bbr 277	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) ∧ (#‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
28		ffn 5958	. . . . . . 7 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → 𝐹 Fn (0..^2))
29		hashfn 13025	. . . . . . . 8 ⊢ (𝐹 Fn (0..^2) → (#‘𝐹) = (#‘(0..^2)))
30		2nn0 11186	. . . . . . . . 9 ⊢ 2 ∈ ℕ0
31		hashfzo0 13077	. . . . . . . . 9 ⊢ (2 ∈ ℕ0 → (#‘(0..^2)) = 2)
32	30, 31	ax-mp 5	. . . . . . . 8 ⊢ (#‘(0..^2)) = 2
33	29, 32	syl6eq 2660	. . . . . . 7 ⊢ (𝐹 Fn (0..^2) → (#‘𝐹) = 2)
34	28, 33	syl 17	. . . . . 6 ⊢ (𝐹:(0..^2)⟶dom 𝐸 → (#‘𝐹) = 2)
35	34	pm4.71i 662	. . . . 5 ⊢ (𝐹:(0..^2)⟶dom 𝐸 ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2))
36	35	bicomi 213	. . . 4 ⊢ ((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐸)
37	36	a1i 11	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → ((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐸))
38	37	3anbi1d 1395	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
39	2, 27, 38	3bitrd 293	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {cpr 4127   class class class wbr 4583  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℕ₀cn0 11169  ...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-2 10956  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:  usg2wlkonot  26410  usg2wotspth  26411