Theorem clwwlknonmpo 16549

Description: (ClWWalksNOn ` G ) is an operator mapping a vertex v and a nonnegative integer n to the set of closed walks on v of length n as words over the set of vertices in a graph G. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)

Assertion

Ref	Expression
clwwlknonmpo	|- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )

Distinct variable group:   n, G, v, w

Proof of Theorem clwwlknonmpo

Dummy variables g s x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clwwlknon 16548	. . . 4 |- ClWWalksNOn = ( g e. _V |-> ( v e. (Vtx ` g ) , n e. NN0 |-> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } ) )
2	1	mptrcl 5765	. . 3 |- ( x e. (ClWWalksNOn ` G ) -> G e. _V )
3		eqid 2234	. . . . 5 |- ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
4	3	elmpom 6447	. . . 4 |- ( x e. ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) -> E. s s e. (Vtx ` G ) )
5		df-vtx 16135	. . . . . 6 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
6	5	mptrcl 5765	. . . . 5 |- ( s e. (Vtx ` G ) -> G e. _V )
7	6	exlimiv 1647	. . . 4 |- ( E. s s e. (Vtx ` G ) -> G e. _V )
8	4, 7	syl 14	. . 3 |- ( x e. ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) -> G e. _V )
9		fveq2 5675	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
10		eqidd 2235	. . . . . 6 |- ( g = G -> NN0 = NN0 )
11		oveq2 6066	. . . . . . 7 |- ( g = G -> ( n ClWWalksN g ) = ( n ClWWalksN G ) )
12	11	rabeqdv 2809	. . . . . 6 |- ( g = G -> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } = { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
13	9, 10, 12	mpoeq123dv 6123	. . . . 5 |- ( g = G -> ( v e. (Vtx ` g ) , n e. NN0 |-> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) )
14		id 19	. . . . 5 |- ( G e. _V -> G e. _V )
15		vtxex 16139	. . . . . 6 |- ( G e. _V -> (Vtx ` G ) e. _V )
16		nn0ex 9519	. . . . . 6 |- NN0 e. _V
17		mpoexga 6421	. . . . . 6 |- ( ( (Vtx ` G ) e. _V /\ NN0 e. _V ) -> ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) e. _V )
18	15, 16, 17	sylancl 413	. . . . 5 |- ( G e. _V -> ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) e. _V )
19	1, 13, 14, 18	fvmptd3 5776	. . . 4 |- ( G e. _V -> (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) )
20	19	eleq2d 2304	. . 3 |- ( G e. _V -> ( x e. (ClWWalksNOn ` G ) <-> x e. ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) ) )
21	2, 8, 20	pm5.21nii 712	. 2 |- ( x e. (ClWWalksNOn ` G ) <-> x e. ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) )
22	21	eqriv 2231	1 |- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )

Syntax hints:   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   _Vcvv 2815   ifcif 3624   X. cxp 4752   ` cfv 5357  (class class class)co 6058   e. cmpo 6060   1stc1st 6345   0cc0 8143   NN0cn0 9513   Basecbs 13296  Vtxcvtx 16133   ClWWalksN cclwwlkn 16524  ClWWalksNOncclwwlknon 16547

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-pow 4292  ax-pr 4327  ax-un 4559  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-i2m1 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  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-id 4419  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-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-inn 9255  df-n0 9514  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135  df-clwwlknon 16548

This theorem is referenced by:  clwwlknon  16550  clwwlk0on0  16552