Theorem clwlksfclwwlk1hashn 26959

Description: The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 2-May-2021.)

Hypotheses

Ref	Expression
clwlksfclwwlk.1	|- A = ( 1st ` c )
clwlksfclwwlk.2	|- B = ( 2nd ` c )
clwlksfclwwlk.c	|- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
clwlksfclwwlk.f	|- F = ( c e. C |-> ( B substr <. 0 , ( # ` A ) >. ) )

Assertion

Ref	Expression
clwlksfclwwlk1hashn	|- ( W e. C -> ( # ` ( 1st ` W ) ) = N )

Distinct variable groups:   G, c   N, c   W, c

Allowed substitution hints:   A( c)   B( c)   C( c)   F( c)

Proof of Theorem clwlksfclwwlk1hashn

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . . . . 6 |- A = ( 1st ` c )
2	1	fveq2i 6194	. . . . 5 |- ( # ` A ) = ( # ` ( 1st ` c ) )
3	2	eqeq1i 2627	. . . 4 |- ( ( # ` A ) = N <-> ( # ` ( 1st ` c ) ) = N )
4		fveq2 6191	. . . . . 6 |- ( c = W -> ( 1st ` c ) = ( 1st ` W ) )
5	4	fveq2d 6195	. . . . 5 |- ( c = W -> ( # ` ( 1st ` c ) ) = ( # ` ( 1st ` W ) ) )
6	5	eqeq1d 2624	. . . 4 |- ( c = W -> ( ( # ` ( 1st ` c ) ) = N <-> ( # ` ( 1st ` W ) ) = N ) )
7	3, 6	syl5bb 272	. . 3 |- ( c = W -> ( ( # ` A ) = N <-> ( # ` ( 1st ` W ) ) = N ) )
8		clwlksfclwwlk.c	. . 3 |- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
9	7, 8	elrab2 3366	. 2 |- ( W e. C <-> ( W e. (ClWalks ` G ) /\ ( # ` ( 1st ` W ) ) = N ) )
10	9	simprbi 480	1 |- ( W e. C -> ( # ` ( 1st ` W ) ) = N )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   <.cop 4183   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   #chash 13117   substr csubstr 13295  ClWalkscclwlks 26666

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-iota 5851  df-fv 5896

This theorem is referenced by:  clwlksf1clwwlklem  26968  clwlksf1clwwlk  26969