Theorem rusgrnumwwlklem 26865

Description: Lemma for rusgrnumwwlk 26870 etc. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 7-May-2021.)

Hypotheses

Ref	Expression
rusgrnumwwlk.v	|- V = (Vtx ` G )
rusgrnumwwlk.l	|- L = ( v e. V , n e. NN0 |-> ( # ` { w e. ( n WWalksN G ) | ( w ` 0 ) = v } ) )

Assertion

Ref	Expression
rusgrnumwwlklem	|- ( ( P e. V /\ N e. NN0 ) -> ( P L N ) = ( # ` { w e. ( N WWalksN G ) | ( w ` 0 ) = P } ) )

Distinct variable groups:   n, G, v, w   n, N, v, w   P, n, v, w   n, V, v, w

Allowed substitution hints:   L( w, v, n)

Proof of Theorem rusgrnumwwlklem

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 |- ( n = N -> ( n WWalksN G ) = ( N WWalksN G ) )
2	1	adantl 482	. . . 4 |- ( ( v = P /\ n = N ) -> ( n WWalksN G ) = ( N WWalksN G ) )
3		eqeq2 2633	. . . . 5 |- ( v = P -> ( ( w ` 0 ) = v <-> ( w ` 0 ) = P ) )
4	3	adantr 481	. . . 4 |- ( ( v = P /\ n = N ) -> ( ( w ` 0 ) = v <-> ( w ` 0 ) = P ) )
5	2, 4	rabeqbidv 3195	. . 3 |- ( ( v = P /\ n = N ) -> { w e. ( n WWalksN G ) | ( w ` 0 ) = v } = { w e. ( N WWalksN G ) | ( w ` 0 ) = P } )
6	5	fveq2d 6195	. 2 |- ( ( v = P /\ n = N ) -> ( # ` { w e. ( n WWalksN G ) | ( w ` 0 ) = v } ) = ( # ` { w e. ( N WWalksN G ) | ( w ` 0 ) = P } ) )
7		rusgrnumwwlk.l	. 2 |- L = ( v e. V , n e. NN0 |-> ( # ` { w e. ( n WWalksN G ) | ( w ` 0 ) = v } ) )
8		fvex 6201	. 2 |- ( # ` { w e. ( N WWalksN G ) | ( w ` 0 ) = P } ) e. _V
9	6, 7, 8	ovmpt2a 6791	1 |- ( ( P e. V /\ N e. NN0 ) -> ( P L N ) = ( # ` { w e. ( N WWalksN G ) | ( w ` 0 ) = P } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   NN0cn0 11292   #chash 13117  Vtxcvtx 25874   WWalksN cwwlksn 26718

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  ax-sep 4781  ax-nul 4789  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

This theorem is referenced by:  rusgrnumwwlkb0  26866  rusgrnumwwlkb1  26867  rusgr0edg  26868  rusgrnumwwlks  26869  rusgrnumwwlkg  26871