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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 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐸𝑃:(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.
StepHypRef Expression
1 iswlk 26048 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
21anbi1d 737 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (#‘𝐹) = 2)))
3 wrdf 13165 . . . . . . . . . 10 (𝐹 ∈ Word dom 𝐸𝐹:(0..^(#‘𝐹))⟶dom 𝐸)
4 oveq2 6557 . . . . . . . . . . 11 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
54feq2d 5944 . . . . . . . . . 10 ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐸𝐹:(0..^2)⟶dom 𝐸))
63, 5syl5ib 233 . . . . . . . . 9 ((#‘𝐹) = 2 → (𝐹 ∈ Word dom 𝐸𝐹:(0..^2)⟶dom 𝐸))
7 iswrdi 13164 . . . . . . . . 9 (𝐹:(0..^2)⟶dom 𝐸𝐹 ∈ Word dom 𝐸)
86, 7impbid1 214 . . . . . . . 8 ((#‘𝐹) = 2 → (𝐹 ∈ Word dom 𝐸𝐹:(0..^2)⟶dom 𝐸))
98adantl 481 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) ∧ (#‘𝐹) = 2) → (𝐹 ∈ Word dom 𝐸𝐹:(0..^2)⟶dom 𝐸))
10 oveq2 6557 . . . . . . . . 9 ((#‘𝐹) = 2 → (0...(#‘𝐹)) = (0...2))
1110feq2d 5944 . . . . . . . 8 ((#‘𝐹) = 2 → (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...2)⟶𝑉))
1211adantl 481 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) ∧ (#‘𝐹) = 2) → (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...2)⟶𝑉))
13 fzo0to2pr 12420 . . . . . . . . . . 11 (0..^2) = {0, 1}
144, 13syl6eq 2660 . . . . . . . . . 10 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
1514raleqdv 3121 . . . . . . . . 9 ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
16 2wlklem 26094 . . . . . . . . 9 (∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
1715, 16syl6bb 275 . . . . . . . 8 ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
1817adantl 481 . . . . . . 7 ((((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) ∧ (#‘𝐹) = 2) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
199, 12, 183anbi123d 1391 . . . . . 6 ((((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) ∧ (#‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐸𝑃:(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 𝐸𝑃:(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)}))))
2119, 20syl6bb 275 . . . . 5 ((((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) ∧ (#‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (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)})))))
2221ex 449 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → ((#‘𝐹) = 2 → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (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)}))))))
2322pm5.32rd 670 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (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))
2624, 25bitri 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))
2723, 26syl6bbr 277 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (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 𝐸𝐹 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)
3230, 31ax-mp 5 . . . . . . . 8 (#‘(0..^2)) = 2
3329, 32syl6eq 2660 . . . . . . 7 (𝐹 Fn (0..^2) → (#‘𝐹) = 2)
3428, 33syl 17 . . . . . 6 (𝐹:(0..^2)⟶dom 𝐸 → (#‘𝐹) = 2)
3534pm4.71i 662 . . . . 5 (𝐹:(0..^2)⟶dom 𝐸 ↔ (𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2))
3635bicomi 213 . . . 4 ((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐸)
3736a1i 11 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹:(0..^2)⟶dom 𝐸 ∧ (#‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐸))
38373anbi1d 1395 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (((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 𝐸𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
392, 27, 383bitrd 293 1 (((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐸𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
 Colors of variables: wff setvar class 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
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