Theorem df2nd2 6239

Description: An alternate possible definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
df2nd2	|- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )

Distinct variable group:   x, y, z

Proof of Theorem df2nd2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo2nd 6177	. . . . 5 |- 2nd : _V -onto-> _V
2		fofn 5455	. . . . 5 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3		dffn5im 5577	. . . . 5 |- ( 2nd Fn _V -> 2nd = ( w e. _V |-> ( 2nd ` w ) ) )
4	1, 2, 3	mp2b 8	. . . 4 |- 2nd = ( w e. _V |-> ( 2nd ` w ) )
5		mptv 4115	. . . 4 |- ( w e. _V |-> ( 2nd ` w ) ) = { <. w , z >. | z = ( 2nd ` w ) }
6	4, 5	eqtri 2210	. . 3 |- 2nd = { <. w , z >. | z = ( 2nd ` w ) }
7	6	reseq1i 4918	. 2 |- ( 2nd |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) )
8		resopab 4966	. 2 |- ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) }
9		vex 2755	. . . . 5 |- x e. _V
10		vex 2755	. . . . 5 |- y e. _V
11	9, 10	op2ndd 6168	. . . 4 |- ( w = <. x , y >. -> ( 2nd ` w ) = y )
12	11	eqeq2d 2201	. . 3 |- ( w = <. x , y >. -> ( z = ( 2nd ` w ) <-> z = y ) )
13	12	dfoprab3 6210	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) } = { <. <. x , y >. , z >. | z = y }
14	7, 8, 13	3eqtrri 2215	1 |- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )

Syntax hints:   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   <.cop 3610   {copab 4078   |-> cmpt 4079   X. cxp 4639   |` cres 4643   Fn wfn 5226   -onto->wfo 5229   ` cfv 5231   {coprab 5892   2ndc2nd 6158

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fo 5237  df-fv 5239  df-oprab 5895  df-1st 6159  df-2nd 6160

This theorem is referenced by: (None)