Theorem wlk2f 16148

Description: If there is a walk 𝑊 there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

Assertion

Ref	Expression
wlk2f	⊢ (𝑊 ∈ (Walks‘𝐺) → ∃𝑓∃𝑝 𝑓(Walks‘𝐺)𝑝)

Distinct variable groups:   𝑓,𝐺,𝑝   𝑓,𝑊,𝑝

Proof of Theorem wlk2f

Step	Hyp	Ref	Expression
1		1stexg 6325	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊) ∈ V)
2		2ndexg 6326	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2nd ‘𝑊) ∈ V)
3	1, 2	jca 306	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((1st ‘𝑊) ∈ V ∧ (2nd ‘𝑊) ∈ V))
4		wlkcprim 16147	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))
5		breq12 4091	. . 3 ⊢ ((𝑓 = (1st ‘𝑊) ∧ 𝑝 = (2nd ‘𝑊)) → (𝑓(Walks‘𝐺)𝑝 ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊)))
6	5	spc2egv 2894	. 2 ⊢ (((1st ‘𝑊) ∈ V ∧ (2nd ‘𝑊) ∈ V) → ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) → ∃𝑓∃𝑝 𝑓(Walks‘𝐺)𝑝))
7	3, 4, 6	sylc 62	1 ⊢ (𝑊 ∈ (Walks‘𝐺) → ∃𝑓∃𝑝 𝑓(Walks‘𝐺)𝑝)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538   ∈ wcel 2200  Vcvv 2800   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-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-1st 6298  df-2nd 6299  df-wlks 16115

This theorem is referenced by: (None)