Theorem iswspthn 26736

Description: An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)

Assertion

Ref	Expression
iswspthn	|- ( W e. ( N WSPathsN G ) <-> ( 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 iswspthn

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4657	. . 3 |- ( w = W -> ( f (SPaths ` G ) w <-> f (SPaths ` G ) W ) )
2	1	exbidv 1850	. 2 |- ( w = W -> ( E. f f (SPaths ` G ) w <-> E. f f (SPaths ` G ) W ) )
3		wspthsn 26735	. 2 |- ( N WSPathsN G ) = { w e. ( N WWalksN G ) | E. f f (SPaths ` G ) w }
4	2, 3	elrab2 3366	1 |- ( W e. ( N WSPathsN G ) <-> ( W e. ( N WWalksN G ) /\ E. f f (SPaths ` G ) W ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  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-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:  wspthnp  26737  wspthsnwspthsnon  26811