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Theorem 2nd2val 6313
Description: Value of an alternate definition of the  2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.)
Assertion
Ref Expression
2nd2val  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
)
Distinct variable group:    x, y, z
Allowed substitution hints:    A( x, y, z)

Proof of Theorem 2nd2val
Dummy variables  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4877 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5669 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. ) )
3 df-ov 6024 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )
4 vex 2903 . . . . . . . 8  |-  w  e. 
_V
5 vex 2903 . . . . . . . 8  |-  v  e. 
_V
6 simpr 448 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  y  =  v )
7 mpt2v 6103 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  y }
87eqcomi 2392 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( x  e.  _V ,  y  e.  _V  |->  y )
96, 8, 5ovmpt2a 6144 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  y } v )  =  v )
104, 5, 9mp2an 654 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  v
113, 10eqtr3i 2410 . . . . . 6  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )  =  v
122, 11syl6eq 2436 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  v )
134, 5op2ndd 6298 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 2nd `  A
)  =  v )
1412, 13eqtr4d 2423 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
1514exlimivv 1642 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
161, 15sylbi 188 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
17 vex 2903 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 2903 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 442 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 a9ev 1663 . . . . . . . . 9  |-  E. z 
z  =  y
2119, 202th 231 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  y )
2221opabbii 4214 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  y }
23 df-xp 4825 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 6094 . . . . . . 7  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  { <. x ,  y >.  |  E. z  z  =  y }
2522, 23, 243eqtr4ri 2419 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  ( _V  X.  _V )
2625eleq2i 2452 . . . . 5  |-  ( A  e.  dom  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  <->  A  e.  ( _V  X.  _V )
)
27 ndmfv 5696 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  (/) )
2826, 27sylnbir 299 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  (/) )
29 rnsnn0 5277 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
3029biimpri 198 . . . . . . 7  |-  ( ran 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2594 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ran  { A }  =  (/) )
3231unieqd 3969 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  U. (/) )
33 uni0 3985 . . . . 5  |-  U. (/)  =  (/)
3432, 33syl6eq 2436 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  (/) )
3528, 34eqtr4d 2423 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  = 
U. ran  { A } )
36 2ndval 6292 . . 3  |-  ( 2nd `  A )  =  U. ran  { A }
3735, 36syl6eqr 2438 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
3816, 37pm2.61i 158 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900   (/)c0 3572   {csn 3758   <.cop 3761   U.cuni 3958   {copab 4207    X. cxp 4817   dom cdm 4819   ran crn 4820   ` cfv 5395  (class class class)co 6021   {coprab 6022    e. cmpt2 6023   2ndc2nd 6288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-2nd 6290
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