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Theorem wlkcpr 26933
Description: A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
Assertion
Ref Expression
wlkcpr (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))

Proof of Theorem wlkcpr
StepHypRef Expression
1 wlkop 26932 . 2 (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
2 wlkvv 26931 . . 3 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → 𝑊 ∈ (V × V))
3 1st2ndb 7473 . . 3 (𝑊 ∈ (V × V) ↔ 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
42, 3sylib 210 . 2 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
5 eleq1 2894 . . 3 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ (Walks‘𝐺)))
6 df-br 4876 . . 3 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ (Walks‘𝐺))
75, 6syl6bbr 281 . 2 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊)))
81, 4, 7pm5.21nii 370 1 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1656  wcel 2164  Vcvv 3414  cop 4405   class class class wbr 4875   × cxp 5344  cfv 6127  1st c1st 7431  2nd c2nd 7432  Walkscwlks 26901
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-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-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-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-n0 11626  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-wlks 26904
This theorem is referenced by:  wlk2f  26934  wlkcompim  26936  upgrwlkcompim  26947  uspgr2wlkeqi  26952  wlkv0  26955  g0wlk0  26956  wlkswwlksf1o  27185  wlknewwlksn  27194  wlknwwlksnbij  27199  clwlkclwwlkf1OLD  27350  clwlkclwwlkf1  27354  clwlknf1oclwwlknlem1  27447  clwlknf1oclwwlknlem1OLD  27448  clwlknf1oclwwlkn  27451  clwlknf1oclwwlknOLD  27453  clwwlknonclwlknonf1o  27756  clwwlknonclwlknonf1oOLD  27757  dlwwlknondlwlknonf1olem1  27760  dlwwlknonclwlknonf1olem1OLD  27761
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