Theorem wlk2f 16201

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 e. (Walks ` G ) -> E. f E. p f (Walks ` G ) p )

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

Proof of Theorem wlk2f

Step	Hyp	Ref	Expression
1		1stexg 6329	. . 3 |- ( W e. (Walks ` G ) -> ( 1st ` W ) e. _V )
2		2ndexg 6330	. . 3 |- ( W e. (Walks ` G ) -> ( 2nd ` W ) e. _V )
3	1, 2	jca 306	. 2 |- ( W e. (Walks ` G ) -> ( ( 1st ` W ) e. _V /\ ( 2nd ` W ) e. _V ) )
4		wlkcprim 16200	. 2 |- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
5		breq12 4093	. . 3 |- ( ( f = ( 1st ` W ) /\ p = ( 2nd ` W ) ) -> ( f (Walks ` G ) p <-> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) )
6	5	spc2egv 2896	. 2 |- ( ( ( 1st ` W ) e. _V /\ ( 2nd ` W ) e. _V ) -> ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> E. f E. p f (Walks ` G ) p ) )
7	3, 4, 6	sylc 62	1 |- ( W e. (Walks ` G ) -> E. f E. p f (Walks ` G ) p )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1540   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   ` cfv 5326   1stc1st 6300   2ndc2nd 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-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6302  df-2nd 6303  df-wlks 16168

This theorem is referenced by: (None)