Theorem wlkcprim 16200

Description: A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Revised by Jim Kingdon, 1-Feb-2026.)

Assertion

Ref	Expression
wlkcprim	⊢ (𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))

Proof of Theorem wlkcprim

Step	Hyp	Ref	Expression
1		wlkop 16198	. . . 4 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
2	1	eleq1d 2300	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝑊 ∈ (Walks‘𝐺) ↔ ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (Walks‘𝐺)))
3	2	ibi 176	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (Walks‘𝐺))
4		df-br 4089	. 2 ⊢ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) ↔ ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (Walks‘𝐺))
5	3, 4	sylibr 134	1 ⊢ (𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))

Syntax hints:   → wi 4   ∈ wcel 2202  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  1^st c1st 6300  2^nd c2nd 6301  Walkscwlks 16167

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302  df-2nd 6303  df-wlks 16168

This theorem is referenced by:  wlk2f  16201  wlkcompim  16202  wlkeq  16204  upgrwlkcompim  16212  uspgr2wlkeqi  16217  wlkv0  16219  g0wlk0  16220