Theorem wlk2f 16062

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 6313	. . 3 |- ( W e. (Walks ` G ) -> ( 1st ` W ) e. _V )
2		2ndexg 6314	. . 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 16061	. 2 |- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
5		breq12 4088	. . 3 |- ( ( f = ( 1st ` W ) /\ p = ( 2nd ` W ) ) -> ( f (Walks ` G ) p <-> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) )
6	5	spc2egv 2893	. 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 1538   e. wcel 2200   _Vcvv 2799   class class class wbr 4083   ` cfv 5318   1stc1st 6284   2ndc2nd 6285  Walkscwlks 16030

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

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 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-1st 6286  df-2nd 6287  df-wlks 16031

This theorem is referenced by: (None)