MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df2nd2 Unicode version

Theorem df2nd2 6290
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 6224 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5533 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 8 . . . . 5  |-  2nd  Fn  _V
4 dffn5 5648 . . . . 5  |-  ( 2nd 
Fn  _V  <->  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) ) )
53, 4mpbi 199 . . . 4  |-  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) )
6 mptv 4191 . . . 4  |-  ( w  e.  _V  |->  ( 2nd `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
75, 6eqtri 2378 . . 3  |-  2nd  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
87reseq1i 5030 . 2  |-  ( 2nd  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )
9 resopab 5075 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 2nd `  w ) ) }
10 vex 2867 . . . . 5  |-  x  e. 
_V
11 vex 2867 . . . . 5  |-  y  e. 
_V
1210, 11op2ndd 6215 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 2nd `  w
)  =  y )
1312eqeq2d 2369 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 2nd `  w
)  <->  z  =  y ) )
1413dfoprab3 6260 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 2nd `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  y }
158, 9, 143eqtrri 2383 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719   {copab 4155    e. cmpt 4156    X. cxp 4766    |` cres 4770    Fn wfn 5329   -onto->wfo 5332   ` cfv 5334   {coprab 5943   2ndc2nd 6205
This theorem is referenced by:  df2ndres  23293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fo 5340  df-fv 5342  df-oprab 5946  df-1st 6206  df-2nd 6207
  Copyright terms: Public domain W3C validator