Theorem clwwlkng 16526

Description: The set of closed walks of a fixed length N as words over the set of vertices in a graph G. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)

Assertion

Ref	Expression
clwwlkng	|- ( ( N e. NN0 /\ G e. V ) -> ( N ClWWalksN G ) = { w e. (ClWWalks ` G ) | (♯ ` w ) = N } )

Distinct variable groups:   w, G   w, N

Allowed substitution hint:   V( w)

Proof of Theorem clwwlkng

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

Step	Hyp	Ref	Expression
1		simpl 109	. 2 |- ( ( N e. NN0 /\ G e. V ) -> N e. NN0 )
2		elex 2827	. . 3 |- ( G e. V -> G e. _V )
3	2	adantl 277	. 2 |- ( ( N e. NN0 /\ G e. V ) -> G e. _V )
4		eqid 2234	. . 3 |- { w e. (ClWWalks ` G ) | (♯ ` w ) = N } = { w e. (ClWWalks ` G ) | (♯ ` w ) = N }
5		clwwlkex 16519	. . . 4 |- ( G e. V -> (ClWWalks ` G ) e. _V )
6	5	adantl 277	. . 3 |- ( ( N e. NN0 /\ G e. V ) -> (ClWWalks ` G ) e. _V )
7	4, 6	rabexd 4262	. 2 |- ( ( N e. NN0 /\ G e. V ) -> { w e. (ClWWalks ` G ) | (♯ ` w ) = N } e. _V )
8		fveq2 5675	. . . . 5 |- ( g = G -> (ClWWalks ` g ) = (ClWWalks ` G ) )
9	8	adantl 277	. . . 4 |- ( ( n = N /\ g = G ) -> (ClWWalks ` g ) = (ClWWalks ` G ) )
10		eqeq2 2244	. . . . 5 |- ( n = N -> ( (♯ ` w ) = n <-> (♯ ` w ) = N ) )
11	10	adantr 276	. . . 4 |- ( ( n = N /\ g = G ) -> ( (♯ ` w ) = n <-> (♯ ` w ) = N ) )
12	9, 11	rabeqbidv 2810	. . 3 |- ( ( n = N /\ g = G ) -> { w e. (ClWWalks ` g ) | (♯ ` w ) = n } = { w e. (ClWWalks ` G ) | (♯ ` w ) = N } )
13		df-clwwlkn 16525	. . 3 |- ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. (ClWWalks ` g ) | (♯ ` w ) = n } )
14	12, 13	ovmpoga 6191	. 2 |- ( ( N e. NN0 /\ G e. _V /\ { w e. (ClWWalks ` G ) | (♯ ` w ) = N } e. _V ) -> ( N ClWWalksN G ) = { w e. (ClWWalks ` G ) | (♯ ` w ) = N } )
15	1, 3, 7, 14	syl3anc 1274	1 |- ( ( N e. NN0 /\ G e. V ) -> ( N ClWWalksN G ) = { w e. (ClWWalks ` G ) | (♯ ` w ) = N } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   ` cfv 5357  (class class class)co 6058   NN0cn0 9513  ♯chash 11163  ClWWalkscclwwlk 16512   ClWWalksN cclwwlkn 16524

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-in1 619  ax-in2 620  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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135  df-clwwlk 16513  df-clwwlkn 16525

This theorem is referenced by:  isclwwlkng  16527  clwwlkn0  16529