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Theorem df2nd2 6860
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 6795 . . . . . 6  |-  2nd : _V -onto-> _V
2 fofn 5788 . . . . . 6  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . . . 5  |-  2nd  Fn  _V
4 dffn5 5904 . . . . 5  |-  ( 2nd 
Fn  _V  <->  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) ) )
53, 4mpbi 208 . . . 4  |-  2nd  =  ( w  e.  _V  |->  ( 2nd `  w ) )
6 mptv 4532 . . . 4  |-  ( w  e.  _V  |->  ( 2nd `  w ) )  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
75, 6eqtri 2489 . . 3  |-  2nd  =  { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }
87reseq1i 5260 . 2  |-  ( 2nd  |`  ( _V  X.  _V ) )  =  ( { <. w ,  z
>.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )
9 resopab 5311 . 2  |-  ( {
<. w ,  z >.  |  z  =  ( 2nd `  w ) }  |`  ( _V  X.  _V ) )  =  { <. w ,  z >.  |  ( w  e.  ( _V  X.  _V )  /\  z  =  ( 2nd `  w ) ) }
10 vex 3109 . . . . 5  |-  x  e. 
_V
11 vex 3109 . . . . 5  |-  y  e. 
_V
1210, 11op2ndd 6785 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( 2nd `  w
)  =  y )
1312eqeq2d 2474 . . 3  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( 2nd `  w
)  <->  z  =  y ) )
1413dfoprab3 6830 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( _V 
X.  _V )  /\  z  =  ( 2nd `  w
) ) }  =  { <. <. x ,  y
>. ,  z >.  |  z  =  y }
158, 9, 143eqtrri 2494 1  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   <.cop 4026   {copab 4497    |-> cmpt 4498    X. cxp 4990    |` cres 4994    Fn wfn 5574   -onto->wfo 5577   ` cfv 5579   {coprab 6276   2ndc2nd 6773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-oprab 6279  df-1st 6774  df-2nd 6775
This theorem is referenced by:  df2ndres  27181
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