Theorem wlkelvv 16060

Description: A walk is an ordered pair. (Contributed by Jim Kingdon, 2-Feb-2026.)

Assertion

Ref	Expression
wlkelvv	⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 ∈ (V × V))

Proof of Theorem wlkelvv

Step	Hyp	Ref	Expression
1		wlkop 16059	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
2		1stexg 6313	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊) ∈ V)
3		2ndexg 6314	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2nd ‘𝑊) ∈ V)
4	2, 3	opelxpd 4752	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (V × V))
5	1, 4	eqeltrd 2306	1 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 ∈ (V × V))

Syntax hints:   → wi 4   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669   × cxp 4717  ‘cfv 5318  1^st c1st 6284  2^nd c2nd 6285  Walkscwlks 16030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-1st 6286  df-2nd 6287  df-wlks 16031

This theorem is referenced by:  wlkeq  16065