Theorem df2nd2 6394

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 6330	. . . . 5 |- 2nd : _V -onto-> _V
2		fofn 5570	. . . . 5 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
3		dffn5im 5700	. . . . 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 4191	. . . 4 |- ( w e. _V |-> ( 2nd ` w ) ) = { <. w , z >. | z = ( 2nd ` w ) }
6	4, 5	eqtri 2252	. . 3 |- 2nd = { <. w , z >. | z = ( 2nd ` w ) }
7	6	reseq1i 5015	. 2 |- ( 2nd |` ( _V X. _V ) ) = ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) )
8		resopab 5063	. 2 |- ( { <. w , z >. | z = ( 2nd ` w ) } |` ( _V X. _V ) ) = { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) }
9		vex 2806	. . . . 5 |- x e. _V
10		vex 2806	. . . . 5 |- y e. _V
11	9, 10	op2ndd 6321	. . . 4 |- ( w = <. x , y >. -> ( 2nd ` w ) = y )
12	11	eqeq2d 2243	. . 3 |- ( w = <. x , y >. -> ( z = ( 2nd ` w ) <-> z = y ) )
13	12	dfoprab3 6363	. 2 |- { <. w , z >. | ( w e. ( _V X. _V ) /\ z = ( 2nd ` w ) ) } = { <. <. x , y >. , z >. | z = y }
14	7, 8, 13	3eqtrri 2257	1 |- { <. <. x , y >. , z >. | z = y } = ( 2nd |` ( _V X. _V ) )

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   <.cop 3676   {copab 4154   |-> cmpt 4155   X. cxp 4729   |` cres 4733   Fn wfn 5328   -onto->wfo 5331   ` cfv 5333   {coprab 6029   2ndc2nd 6311

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 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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-oprab 6032  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)