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Theorem wlkcpr 29576

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 29575	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2		wlkvv 29574	. . 3 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → 𝑊 ∈ (V × V))
3		1st2ndb 8036	. . 3 ⊢ (𝑊 ∈ (V × V) ↔ 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
4	2, 3	sylib 218	. 2 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
5		eleq1 2821	. . 3 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (Walks‘𝐺)))
6		df-br 5124	. . 3 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (Walks‘𝐺))
7	5, 6	bitr4di 289	. 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 378	1 ⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1539 ∈ wcel 2107 Vcvv 3463 ⟨cop 4612 class class class wbr 5123 × cxp 5663 ‘cfv 6541 1^st c1st 7994 2^nd c2nd 7995 Walkscwlks 29543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-card 9961 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-n0 12510 df-z 12597 df-uz 12861 df-fz 13530 df-fzo 13677 df-hash 14353 df-word 14536 df-wlks 29546

This theorem is referenced by: wlk2f 29577 wlkcompim 29579 upgrwlkcompim 29590 uspgr2wlkeqi 29595 wlkv0 29598 g0wlk0 29599 wlkswwlksf1o 29828 wlknewwlksn 29836 wlknwwlksnbij 29837 clwlkclwwlkf1 29958 clwlknf1oclwwlknlem1 30029 clwlknf1oclwwlkn 30032 clwwlknonclwlknonf1o 30310 dlwwlknondlwlknonf1olem1 30312

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