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Theorem df2nd2 6437
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.
StepHypRef Expression
1 fo2nd 6370 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5658 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  2nd  Fn  _V
4 dffn5 5775 . . . . 5  |-  ( 2nd 
Fn  _V  <->  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) ) )
53, 4mpbi 201 . . . 4  |-  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) )
6 mptv 4304 . . . 4  |-  ( w  e.  _V  |->  ( 2nd `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
75, 6eqtri 2458 . . 3  |-  2nd  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
87reseq1i 5145 . 2  |-  ( 2nd  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )
9 resopab 5190 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 2nd `  w ) ) }
10 vex 2961 . . . . 5  |-  x  e. 
_V
11 vex 2961 . . . . 5  |-  y  e. 
_V
1210, 11op2ndd 6361 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 2nd `  w
)  =  y )
1312eqeq2d 2449 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 2nd `  w
)  <->  z  =  y ) )
1413dfoprab3 6406 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 2nd `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  y }
158, 9, 143eqtrri 2463 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   {copab 4268    e. cmpt 4269    X. cxp 4879    |` cres 4883    Fn wfn 5452   -onto->wfo 5455   ` cfv 5457   {coprab 6085   2ndc2nd 6351
This theorem is referenced by:  df2ndres  24097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-oprab 6088  df-1st 6352  df-2nd 6353
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