Theorem wspthnp 26737

Description: Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.)

Assertion

Ref	Expression
wspthnp	|- ( W e. ( N WSPathsN G ) -> ( ( N e. NN0 /\ G e. _V ) /\ W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) )

Distinct variable groups:   f, G   f, W

Allowed substitution hint:   N( f)

Proof of Theorem wspthnp

Dummy variables g n w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-wspthsn 26725	. . 3 |- WSPathsN = ( n e. NN0 , g e. _V |-> { w e. ( n WWalksN g ) | E. f f (SPaths ` g ) w } )
2	1	elmpt2cl 6876	. 2 |- ( W e. ( N WSPathsN G ) -> ( N e. NN0 /\ G e. _V ) )
3		simpl 473	. . 3 |- ( ( ( N e. NN0 /\ G e. _V ) /\ W e. ( N WSPathsN G ) ) -> ( N e. NN0 /\ G e. _V ) )
4		iswspthn 26736	. . . . 5 |- ( W e. ( N WSPathsN G ) <-> ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) )
5	4	a1i 11	. . . 4 |- ( ( N e. NN0 /\ G e. _V ) -> ( W e. ( N WSPathsN G ) <-> ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) ) )
6	5	biimpa 501	. . 3 |- ( ( ( N e. NN0 /\ G e. _V ) /\ W e. ( N WSPathsN G ) ) -> ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) )
7		3anass 1042	. . 3 |- ( ( ( N e. NN0 /\ G e. _V ) /\ W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) <-> ( ( N e. NN0 /\ G e. _V ) /\ ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) ) )
8	3, 6, 7	sylanbrc 698	. 2 |- ( ( ( N e. NN0 /\ G e. _V ) /\ W e. ( N WSPathsN G ) ) -> ( ( N e. NN0 /\ G e. _V ) /\ W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) )
9	2, 8	mpancom 703	1 |- ( W e. ( N WSPathsN G ) -> ( ( N e. NN0 /\ G e. _V ) /\ W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   NN0cn0 11292  SPathscspths 26609   WWalksN cwwlksn 26718   WSPathsN cwwspthsn 26720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-wwlksn 26723  df-wspthsn 26725

This theorem is referenced by:  wspthsswwlkn  26814