Theorem clwwlknonmpo 16213

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 16212	. . . 4 |- ClWWalksNOn = ( g e. _V |-> ( v e. (Vtx ` g ) , n e. NN0 |-> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } ) )
2	1	mptrcl 5723	. . 3 |- ( x e. (ClWWalksNOn ` G ) -> G e. _V )
3		eqid 2229	. . . . 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 6396	. . . 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 15852	. . . . . 6 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
6	5	mptrcl 5723	. . . . 5 |- ( s e. (Vtx ` G ) -> G e. _V )
7	6	exlimiv 1644	. . . 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 5633	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
10		eqidd 2230	. . . . . 6 |- ( g = G -> NN0 = NN0 )
11		oveq2 6019	. . . . . . 7 |- ( g = G -> ( n ClWWalksN g ) = ( n ClWWalksN G ) )
12	11	rabeqdv 2794	. . . . . 6 |- ( g = G -> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } = { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
13	9, 10, 12	mpoeq123dv 6076	. . . . 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 15856	. . . . . 6 |- ( G e. _V -> (Vtx ` G ) e. _V )
16		nn0ex 9396	. . . . . 6 |- NN0 e. _V
17		mpoexga 6370	. . . . . 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 5734	. . . 4 |- ( G e. _V -> (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) )
20	19	eleq2d 2299	. . 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 709	. 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 2226	1 |- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )

Syntax hints:   = wceq 1395   E.wex 1538   e. wcel 2200   {crab 2512   _Vcvv 2800   ifcif 3603   X. cxp 4719   ` cfv 5322  (class class class)co 6011   e. cmpo 6013   1stc1st 6294   0cc0 8020   NN0cn0 9390   Basecbs 13069  Vtxcvtx 15850   ClWWalksN cclwwlkn 16188  ClWWalksNOncclwwlknon 16211

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-i2m1 8125

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-inn 9132  df-n0 9391  df-ndx 13072  df-slot 13073  df-base 13075  df-vtx 15852  df-clwwlknon 16212

This theorem is referenced by:  clwwlknon  16214  clwwlk0on0  16216