wlkcpr - Metamath Proof Explorer

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Theorem wlkcpr 29989

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 29988	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩)
2		wlkvv 29987	. . 3 ⊢ ((1^st ‘𝑊)(Walks‘𝐺)(2^nd ‘𝑊) → 𝑊 ∈ (V × V))
3		1st2ndb 8024	. . 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 2850	. . 3 ⊢ (𝑊 = ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨(1^st ‘𝑊), (2^nd ‘𝑊)⟩ ∈ (Walks‘𝐺)))
6		df-br 5109	. . 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 2142 Vcvv 3454 ⟨cop 4594 class class class wbr 5108 × cxp 5658 ‘cfv 6536 1^st c1st 7982 2^nd c2nd 7983 Walkscwlks 29957

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1078 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-n0 12511 df-z 12598 df-uz 12869 df-fz 13542 df-fzo 13690 df-hash 14374 df-word 14558 df-wlks 29960

This theorem is used by: wlk2f 29990 wlkcompim 29992 upgrwlkcompim 30003 uspgr2wlkeqi 30008 wlkv0 30010 g0wlk0 30011 wlkswwlksf1o 30239 wlknewwlksn 30247 wlknwwlksnbij 30248 clwlkclwwlkf1 30372 clwlknf1oclwwlknlem1 30443 clwlknf1oclwwlkn 30446 clwwlknonclwlknonf1o 30724 dlwwlknondlwlknonf1olem1 30726

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