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Theorem df2nd2 5872
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 5816 . . . . 5  |-  2nd : _V -onto-> _V
2 fofn 5139 . . . . 5  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
3 dffn5im 5251 . . . . 5  |-  ( 2nd 
Fn  _V  ->  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) ) )
41, 2, 3mp2b 8 . . . 4  |-  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) )
5 mptv 3882 . . . 4  |-  ( w  e.  _V  |->  ( 2nd `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
64, 5eqtri 2102 . . 3  |-  2nd  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
76reseq1i 4636 . 2  |-  ( 2nd  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )
8 resopab 4682 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 2nd `  w ) ) }
9 vex 2605 . . . . 5  |-  x  e. 
_V
10 vex 2605 . . . . 5  |-  y  e. 
_V
119, 10op2ndd 5807 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 2nd `  w
)  =  y )
1211eqeq2d 2093 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 2nd `  w
)  <->  z  =  y ) )
1312dfoprab3 5848 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 2nd `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  y }
147, 8, 133eqtrri 2107 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   <.cop 3409   {copab 3846    |-> cmpt 3847    X. cxp 4369    |` cres 4373    Fn wfn 4927   -onto->wfo 4930   ` cfv 4932   {coprab 5544   2ndc2nd 5797
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940  df-oprab 5547  df-1st 5798  df-2nd 5799
This theorem is referenced by: (None)
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