Theorem df2nd2 5861

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 5805	. . . . 5 |- 2nd : _V -onto-> _V
2		fofn 5128	. . . . 5 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3		dffn5im 5240	. . . . 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 3874	. . . 4 |- ( w e. _V |-> ( 2nd ` w ) ) = { <. w , z >. | z = ( 2nd ` w ) }
6	4, 5	eqtri 2101	. . 3 |- 2nd = { <. w , z >. | z = ( 2nd ` w ) }
7	6	reseq1i 4626	. 2 |- ( 2nd |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) )
8		resopab 4672	. 2 |- ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) }
9		vex 2604	. . . . 5 |- x e. _V
10		vex 2604	. . . . 5 |- y e. _V
11	9, 10	op2ndd 5796	. . . 4 |- ( w = <. x , y >. -> ( 2nd ` w ) = y )
12	11	eqeq2d 2092	. . 3 |- ( w = <. x , y >. -> ( z = ( 2nd ` w ) <-> z = y ) )
13	12	dfoprab3 5837	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) } = { <. <. x , y >. , z >. | z = y }
14	7, 8, 13	3eqtrri 2106	1 |- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )

Syntax hints:   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   {copab 3838   |-> cmpt 3839   X. cxp 4361   |` cres 4365   Fn wfn 4917   -onto->wfo 4920   ` cfv 4922   {coprab 5533   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-oprab 5536  df-1st 5787  df-2nd 5788

This theorem is referenced by: (None)