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Theorem wlkiswwlkfun 26245
 Description: Lemma 1 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)
Hypotheses
Ref Expression
wlkiswwlkbij.t 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
wlkiswwlkbij.w 𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
wlkiswwlkbij.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlkiswwlkfun ((𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
Distinct variable groups:   𝐸,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇   𝑉,𝑝,𝑡,𝑤   𝑡,𝑊
Allowed substitution hints:   𝑇(𝑤,𝑝)   𝐹(𝑤,𝑡,𝑝)   𝑊(𝑤,𝑝)

Proof of Theorem wlkiswwlkfun
StepHypRef Expression
1 fveq2 6103 . . . . . . . 8 (𝑝 = 𝑡 → (1st𝑝) = (1st𝑡))
21fveq2d 6107 . . . . . . 7 (𝑝 = 𝑡 → (#‘(1st𝑝)) = (#‘(1st𝑡)))
32eqeq1d 2612 . . . . . 6 (𝑝 = 𝑡 → ((#‘(1st𝑝)) = 𝑁 ↔ (#‘(1st𝑡)) = 𝑁))
4 fveq2 6103 . . . . . . . 8 (𝑝 = 𝑡 → (2nd𝑝) = (2nd𝑡))
54fveq1d 6105 . . . . . . 7 (𝑝 = 𝑡 → ((2nd𝑝)‘0) = ((2nd𝑡)‘0))
65eqeq1d 2612 . . . . . 6 (𝑝 = 𝑡 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑡)‘0) = 𝑃))
73, 6anbi12d 743 . . . . 5 (𝑝 = 𝑡 → (((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)))
8 wlkiswwlkbij.t . . . . 5 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
97, 8elrab2 3333 . . . 4 (𝑡𝑇 ↔ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)))
10 simpr 476 . . . . . 6 ((𝑃𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
11 simpl 472 . . . . . . 7 (((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃) → (#‘(1st𝑡)) = 𝑁)
1211anim2i 591 . . . . . 6 ((𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)) → (𝑡 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑡)) = 𝑁))
13 vfwlkniswwlkn 26234 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑡)) = 𝑁)) → (2nd𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
1410, 12, 13syl2an 493 . . . . 5 (((𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → (2nd𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
15 simprrr 801 . . . . 5 (((𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → ((2nd𝑡)‘0) = 𝑃)
1614, 15jca 553 . . . 4 (((𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → ((2nd𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((2nd𝑡)‘0) = 𝑃))
179, 16sylan2b 491 . . 3 (((𝑃𝑉𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → ((2nd𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((2nd𝑡)‘0) = 𝑃))
18 fveq1 6102 . . . . 5 (𝑤 = (2nd𝑡) → (𝑤‘0) = ((2nd𝑡)‘0))
1918eqeq1d 2612 . . . 4 (𝑤 = (2nd𝑡) → ((𝑤‘0) = 𝑃 ↔ ((2nd𝑡)‘0) = 𝑃))
20 wlkiswwlkbij.w . . . 4 𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
2119, 20elrab2 3333 . . 3 ((2nd𝑡) ∈ 𝑊 ↔ ((2nd𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((2nd𝑡)‘0) = 𝑃))
2217, 21sylibr 223 . 2 (((𝑃𝑉𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → (2nd𝑡) ∈ 𝑊)
23 wlkiswwlkbij.f . 2 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
2422, 23fmptd 6292 1 ((𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  ℕ0cn0 11169  #chash 12979   Walks cwalk 26026   WWalksN cwwlkn 26206 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  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wlkiswwlkinj  26246  wlkiswwlksur  26247
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