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Theorem 2nd2val 6146
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 4748 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5525 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. ) )
3 df-ov 5861 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )
4 vex 2791 . . . . . . . 8  |-  w  e. 
_V
5 vex 2791 . . . . . . . 8  |-  v  e. 
_V
6 simpr 447 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  y  =  v )
7 mpt2v 5937 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  y }
87eqcomi 2287 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( x  e.  _V ,  y  e.  _V  |->  y )
96, 8, 5ovmpt2a 5978 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  y } v )  =  v )
104, 5, 9mp2an 653 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  v
113, 10eqtr3i 2305 . . . . . 6  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )  =  v
122, 11syl6eq 2331 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  v )
134, 5op2ndd 6131 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 2nd `  A
)  =  v )
1412, 13eqtr4d 2318 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
1514exlimivv 1667 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
161, 15sylbi 187 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
17 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 441 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 a9ev 1637 . . . . . . . . 9  |-  E. z 
z  =  y
2119, 202th 230 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  y )
2221opabbii 4083 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  y }
23 df-xp 4695 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 5928 . . . . . . 7  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  { <. x ,  y >.  |  E. z  z  =  y }
2522, 23, 243eqtr4ri 2314 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  ( _V  X.  _V )
2625eleq2i 2347 . . . . 5  |-  ( A  e.  dom  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  <->  A  e.  ( _V  X.  _V )
)
27 ndmfv 5552 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  (/) )
2826, 27sylnbir 298 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  (/) )
29 rnsnn0 5139 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
3029biimpri 197 . . . . . . 7  |-  ( ran 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2489 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ran  { A }  =  (/) )
3231unieqd 3838 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  U. (/) )
33 uni0 3854 . . . . 5  |-  U. (/)  =  (/)
3432, 33syl6eq 2331 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  (/) )
3528, 34eqtr4d 2318 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  = 
U. ran  { A } )
36 2ndval 6125 . . 3  |-  ( 2nd `  A )  =  U. ran  { A }
3735, 36syl6eqr 2333 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
3816, 37pm2.61i 156 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640   <.cop 3643   U.cuni 3827   {copab 4076    X. cxp 4687   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858   {coprab 5859    e. cmpt2 5860   2ndc2nd 6121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123
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