Theorem wlk2f 16472

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 6374	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊) ∈ V)
2		2ndexg 6375	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → (2nd ‘𝑊) ∈ V)
3	1, 2	jca 306	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → ((1st ‘𝑊) ∈ V ∧ (2nd ‘𝑊) ∈ V))
4		wlkcprim 16471	. 2 ⊢ (𝑊 ∈ (Walks‘𝐺) → (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))
5		breq12 4119	. . 3 ⊢ ((𝑓 = (1st ‘𝑊) ∧ 𝑝 = (2nd ‘𝑊)) → (𝑓(Walks‘𝐺)𝑝 ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊)))
6	5	spc2egv 2909	. 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 1541   ∈ wcel 2205  Vcvv 2815   class class class wbr 4114  ‘cfv 5357  1^st c1st 6345  2^nd c2nd 6346  Walkscwlks 16438

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-1st 6347  df-2nd 6348  df-wlks 16439

This theorem is referenced by: (None)