Theorem wlk2f 16346

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 6361	. . 3 |- ( W e. (Walks ` G ) -> ( 1st ` W ) e. _V )
2		2ndexg 6362	. . 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 16345	. 2 |- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
5		breq12 4114	. . 3 |- ( ( f = ( 1st ` W ) /\ p = ( 2nd ` W ) ) -> ( f (Walks ` G ) p <-> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) ) )
6	5	spc2egv 2907	. 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 2203   _Vcvv 2813   class class class wbr 4109   ` cfv 5352   1stc1st 6332   2ndc2nd 6333  Walkscwlks 16312

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-1st 6334  df-2nd 6335  df-wlks 16313

This theorem is referenced by: (None)