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Theorem wlkcpr 29614

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 29613	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2		wlkvv 29612	. . 3 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → 𝑊 ∈ (V × V))
3		1st2ndb 8033	. . 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 2823	. . 3 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (Walks‘𝐺)))
6		df-br 5125	. . 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 1540 ∈ wcel 2109 Vcvv 3464 ⟨cop 4612 class class class wbr 5124 × cxp 5657 ‘cfv 6536 1^st c1st 7991 2^nd c2nd 7992 Walkscwlks 29581

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

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 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 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 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-hash 14354 df-word 14537 df-wlks 29584

This theorem is referenced by: wlk2f 29615 wlkcompim 29617 upgrwlkcompim 29628 uspgr2wlkeqi 29633 wlkv0 29636 g0wlk0 29637 wlkswwlksf1o 29866 wlknewwlksn 29874 wlknwwlksnbij 29875 clwlkclwwlkf1 29996 clwlknf1oclwwlknlem1 30067 clwlknf1oclwwlkn 30070 clwwlknonclwlknonf1o 30348 dlwwlknondlwlknonf1olem1 30350

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