Theorem wlkcprim 16147

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 16145	. . . 4 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
2	1	eleq1d 2298	. . 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 4087	. 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 2200  ⟨cop 3670   class class class wbr 4086  ‘cfv 5324  1^st c1st 6296  2^nd c2nd 6297  Walkscwlks 16114

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

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 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fv 5332  df-1st 6298  df-2nd 6299  df-wlks 16115

This theorem is referenced by:  wlk2f  16148  wlkcompim  16149  wlkeq  16151  upgrwlkcompim  16159  uspgr2wlkeqi  16164  wlkv0  16166  g0wlk0  16167