Theorem clwwlknonmpo 16669

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 16668	. . . 4 |- ClWWalksNOn = ( g e. _V |-> ( v e. (Vtx ` g ) , n e. NN0 |-> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } ) )
2	1	mptrcl 5788	. . 3 |- ( x e. (ClWWalksNOn ` G ) -> G e. _V )
3		eqid 2238	. . . . 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 6474	. . . 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 16255	. . . . . 6 |- Vtx = ( g e. _V |-> if ( g e. ( _V X. _V ) , ( 1st ` g ) , ( Base ` g ) ) )
6	5	mptrcl 5788	. . . . 5 |- ( s e. (Vtx ` G ) -> G e. _V )
7	6	exlimiv 1651	. . . 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 5695	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
10		eqidd 2239	. . . . . 6 |- ( g = G -> NN0 = NN0 )
11		oveq2 6093	. . . . . . 7 |- ( g = G -> ( n ClWWalksN g ) = ( n ClWWalksN G ) )
12	11	rabeqdv 2815	. . . . . 6 |- ( g = G -> { w e. ( n ClWWalksN g ) | ( w ` 0 ) = v } = { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
13	9, 10, 12	mpoeq123dv 6150	. . . . 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 16259	. . . . . 6 |- ( G e. _V -> (Vtx ` G ) e. _V )
16		nn0ex 9569	. . . . . 6 |- NN0 e. _V
17		mpoexga 6448	. . . . . 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 417	. . . . 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 5799	. . . 4 |- ( G e. _V -> (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } ) )
20	19	eleq2d 2308	. . 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 716	. 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 2235	1 |- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )

This proof depends on syntax axioms:   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   _Vcvv 2821   ifcif 3638   X. cxp 4772   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   0cc0 8179   NN0cn0 9563   Basecbs 13352  Vtxcvtx 16253   ClWWalksN cclwwlkn 16644  ClWWalksNOncclwwlknon 16667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-i2m1 8284

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-inn 9305  df-n0 9564  df-ndx 13355  df-slot 13356  df-base 13358  df-vtx 16255  df-clwwlknon 16668

This theorem is used by:  clwwlknon  16670  clwwlk0on0  16672