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Theorem wlkwwlkfunOLD 27203
 Description: Obsolete as of 2-Aug-2022. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
wlkwwlkbijOLD.t 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
wlkwwlkbijOLD.w 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
wlkwwlkbijOLD.f 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
Assertion
Ref Expression
wlkwwlkfunOLD ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
Distinct variable groups:   𝐺,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇,𝑤   𝑡,𝑉   𝑡,𝑊
Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑤,𝑡,𝑝)   𝑉(𝑤,𝑝)   𝑊(𝑤,𝑝)

Proof of Theorem wlkwwlkfunOLD
StepHypRef Expression
1 fveq2 6437 . . . . . . . 8 (𝑝 = 𝑡 → (1st𝑝) = (1st𝑡))
21fveq2d 6441 . . . . . . 7 (𝑝 = 𝑡 → (♯‘(1st𝑝)) = (♯‘(1st𝑡)))
32eqeq1d 2827 . . . . . 6 (𝑝 = 𝑡 → ((♯‘(1st𝑝)) = 𝑁 ↔ (♯‘(1st𝑡)) = 𝑁))
4 fveq2 6437 . . . . . . . 8 (𝑝 = 𝑡 → (2nd𝑝) = (2nd𝑡))
54fveq1d 6439 . . . . . . 7 (𝑝 = 𝑡 → ((2nd𝑝)‘0) = ((2nd𝑡)‘0))
65eqeq1d 2827 . . . . . 6 (𝑝 = 𝑡 → (((2nd𝑝)‘0) = 𝑃 ↔ ((2nd𝑡)‘0) = 𝑃))
73, 6anbi12d 624 . . . . 5 (𝑝 = 𝑡 → (((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃) ↔ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)))
8 wlkwwlkbijOLD.t . . . . 5 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}
97, 8elrab2 3589 . . . 4 (𝑡𝑇 ↔ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)))
10 simp1 1170 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐺 ∈ UPGraph)
11 simpl 476 . . . . . . 7 ((𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)) → 𝑡 ∈ (Walks‘𝐺))
1210, 11anim12i 606 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → (𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)))
13 simp3 1172 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
14 simprl 787 . . . . . . 7 ((𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃)) → (♯‘(1st𝑡)) = 𝑁)
1513, 14anim12i 606 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑡)) = 𝑁))
16 wlknewwlksn 27194 . . . . . 6 (((𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑡)) = 𝑁)) → (2nd𝑡) ∈ (𝑁 WWalksN 𝐺))
1712, 15, 16syl2anc 579 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → (2nd𝑡) ∈ (𝑁 WWalksN 𝐺))
18 simprrr 800 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → ((2nd𝑡)‘0) = 𝑃)
1917, 18jca 507 . . . 4 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st𝑡)) = 𝑁 ∧ ((2nd𝑡)‘0) = 𝑃))) → ((2nd𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd𝑡)‘0) = 𝑃))
209, 19sylan2b 587 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → ((2nd𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd𝑡)‘0) = 𝑃))
21 fveq1 6436 . . . . 5 (𝑤 = (2nd𝑡) → (𝑤‘0) = ((2nd𝑡)‘0))
2221eqeq1d 2827 . . . 4 (𝑤 = (2nd𝑡) → ((𝑤‘0) = 𝑃 ↔ ((2nd𝑡)‘0) = 𝑃))
23 wlkwwlkbijOLD.w . . . 4 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
2422, 23elrab2 3589 . . 3 ((2nd𝑡) ∈ 𝑊 ↔ ((2nd𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd𝑡)‘0) = 𝑃))
2520, 24sylibr 226 . 2 (((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) ∧ 𝑡𝑇) → (2nd𝑡) ∈ 𝑊)
26 wlkwwlkbijOLD.f . 2 𝐹 = (𝑡𝑇 ↦ (2nd𝑡))
2725, 26fmptd 6638 1 ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  {crab 3121   ↦ cmpt 4954  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  1st c1st 7431  2nd c2nd 7432  0cc0 10259  ℕ0cn0 11625  ♯chash 13417  UPGraphcupgr 26385  Walkscwlks 26901   WWalksN cwwlksn 27132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-edg 26353  df-uhgr 26363  df-upgr 26387  df-wlks 26904  df-wwlks 27136  df-wwlksn 27137 This theorem is referenced by:  wlkwwlkinjOLD  27204  wlkwwlksurOLD  27205
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