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Theorem wlkcpr 26411
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 26410 . 2 (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
2 wlkvv 26409 . . 3 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → 𝑊 ∈ (V × V))
3 1st2ndb 7158 . . 3 (𝑊 ∈ (V × V) ↔ 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
42, 3sylib 208 . 2 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)
5 eleq1 2686 . . 3 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ (Walks‘𝐺)))
6 df-br 4619 . . 3 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) ↔ ⟨(1st𝑊), (2nd𝑊)⟩ ∈ (Walks‘𝐺))
75, 6syl6bbr 278 . 2 (𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊)))
81, 4, 7pm5.21nii 368 1 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  Vcvv 3189  cop 4159   class class class wbr 4618   × cxp 5077  cfv 5852  1st c1st 7118  2nd c2nd 7119  Walkscwlks 26379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-hash 13066  df-word 13246  df-wlks 26382
This theorem is referenced by:  wlk2f  26412  wlkcompim  26414  upgrwlkcompim  26425  uspgr2wlkeqi  26430  wlkv0  26433  g0wlk0  26434  wlkpwwlkf1ouspgr  26651  wlknewwlksn  26659
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