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Theorem df2nd2 5999
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 5943 . . . . 5  |-  2nd : _V -onto-> _V
2 fofn 5248 . . . . 5  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
3 dffn5im 5363 . . . . 5  |-  ( 2nd 
Fn  _V  ->  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) ) )
41, 2, 3mp2b 8 . . . 4  |-  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) )
5 mptv 3941 . . . 4  |-  ( w  e.  _V  |->  ( 2nd `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
64, 5eqtri 2109 . . 3  |-  2nd  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
76reseq1i 4722 . 2  |-  ( 2nd  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )
8 resopab 4769 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 2nd `  w ) ) }
9 vex 2623 . . . . 5  |-  x  e. 
_V
10 vex 2623 . . . . 5  |-  y  e. 
_V
119, 10op2ndd 5934 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 2nd `  w
)  =  y )
1211eqeq2d 2100 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 2nd `  w
)  <->  z  =  y ) )
1312dfoprab3 5975 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 2nd `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  y }
147, 8, 133eqtrri 2114 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1290    e. wcel 1439   _Vcvv 2620   <.cop 3453   {copab 3904    |-> cmpt 3905    X. cxp 4449    |` cres 4453    Fn wfn 5023   -onto->wfo 5026   ` cfv 5028   {coprab 5667   2ndc2nd 5924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fo 5034  df-fv 5036  df-oprab 5670  df-1st 5925  df-2nd 5926
This theorem is referenced by: (None)
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