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 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 6374	. . 3 |- ( W e. (Walks ` G ) -> ( 1st ` W ) e. _V )
2		2ndexg 6375	. . 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 16471	. 2 |- ( W e. (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 ) 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 1541   e. wcel 2205   _Vcvv 2815   class class class wbr 4114   ` cfv 5357   1stc1st 6345   2ndc2nd 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)