wlkcpr - Metamath Proof Explorer

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Theorem wlkcpr 30209

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) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))

**Proof of Theorem wlkcpr**
Step	Hyp	Ref	Expression
1		wlkop 30208	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2		wlkvv 30207	. . 3 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → 𝑊 ∈ (V × V))
3		1st2ndb 8041	. . 3 ⊢ (𝑊 ∈ (V × V) ↔ 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
4	2, 3	sylib 221	. 2 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
5		eleq1 2848	. . 3 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (Walks‘𝐺)))
6		df-br 5103	. . 3 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (Walks‘𝐺))
7	5, 6	bitr4di 292	. 2 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊)))
8	1, 4, 7	pm5.21nii 381	1 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4589 class class class wbr 5102 × cxp 5648 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000 Walkscwlks 30177

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-hash 14475 df-word 14659 df-wlks 30180

This theorem is used by: wlk2f 30210 wlkcompim 30212 upgrwlkcompim 30223 uspgr2wlkeqi 30228 wlkv0 30230 g0wlk0 30231 wlkswwlksf1o 30468 wlknewwlksn 30476 wlknwwlksnbij 30477 clwlkclwwlkf1 30601 clwlknf1oclwwlknlem1 30672 clwlknf1oclwwlkn 30675 clwwlknonclwlknonf1o 30963 dlwwlknondlwlknonf1olem1 30965

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