Theorem wlk2f 16292

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 6339	. . 3 |- ( W e. (Walks ` G ) -> ( 1st ` W ) e. _V )
2		2ndexg 6340	. . 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 16291	. 2 |- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
5		breq12 4098	. . 3 |- ( ( f = ( 1st ` W ) /\ p = ( 2nd ` W ) ) -> ( f (Walks ` G ) p <-> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) )
6	5	spc2egv 2897	. 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 2202   _Vcvv 2803   class class class wbr 4093   ` cfv 5333   1stc1st 6310   2ndc2nd 6311  Walkscwlks 16258

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312  df-2nd 6313  df-wlks 16259

This theorem is referenced by: (None)