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Theorem 2nd2val 6803
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 5052 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. w E. v  A  =  <. w ,  v >. )
2 fveq2 5859 . . . . . 6  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. ) )
3 df-ov 6280 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )
4 vex 3111 . . . . . . . 8  |-  w  e. 
_V
5 vex 3111 . . . . . . . 8  |-  v  e. 
_V
6 simpr 461 . . . . . . . . 9  |-  ( ( x  =  w  /\  y  =  v )  ->  y  =  v )
7 mpt2v 6369 . . . . . . . . . 10  |-  ( x  e.  _V ,  y  e.  _V  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  z  =  y }
87eqcomi 2475 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( x  e.  _V ,  y  e.  _V  |->  y )
96, 8, 5ovmpt2a 6410 . . . . . . . 8  |-  ( ( w  e.  _V  /\  v  e.  _V )  ->  ( w { <. <.
x ,  y >. ,  z >.  |  z  =  y } v )  =  v )
104, 5, 9mp2an 672 . . . . . . 7  |-  ( w { <. <. x ,  y
>. ,  z >.  |  z  =  y } v )  =  v
113, 10eqtr3i 2493 . . . . . 6  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 <. w ,  v
>. )  =  v
122, 11syl6eq 2519 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  v )
134, 5op2ndd 6787 . . . . 5  |-  ( A  =  <. w ,  v
>.  ->  ( 2nd `  A
)  =  v )
1412, 13eqtr4d 2506 . . . 4  |-  ( A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
1514exlimivv 1694 . . 3  |-  ( E. w E. v  A  =  <. w ,  v
>.  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  ( 2nd `  A ) )
161, 15sylbi 195 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
17 vex 3111 . . . . . . . . . 10  |-  x  e. 
_V
18 vex 3111 . . . . . . . . . 10  |-  y  e. 
_V
1917, 18pm3.2i 455 . . . . . . . . 9  |-  ( x  e.  _V  /\  y  e.  _V )
20 ax6ev 1716 . . . . . . . . 9  |-  E. z 
z  =  y
2119, 202th 239 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  <->  E. z  z  =  y )
2221opabbii 4506 . . . . . . 7  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { <. x ,  y >.  |  E. z  z  =  y }
23 df-xp 5000 . . . . . . 7  |-  ( _V 
X.  _V )  =  { <. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  _V ) }
24 dmoprab 6360 . . . . . . 7  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  { <. x ,  y >.  |  E. z  z  =  y }
2522, 23, 243eqtr4ri 2502 . . . . . 6  |-  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  =  ( _V  X.  _V )
2625eleq2i 2540 . . . . 5  |-  ( A  e.  dom  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  <->  A  e.  ( _V  X.  _V )
)
27 ndmfv 5883 . . . . 5  |-  ( -.  A  e.  dom  { <. <. x ,  y
>. ,  z >.  |  z  =  y }  ->  ( { <. <.
x ,  y >. ,  z >.  |  z  =  y } `  A )  =  (/) )
2826, 27sylnbir 307 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  (/) )
29 rnsnn0 5467 . . . . . . . 8  |-  ( A  e.  ( _V  X.  _V )  <->  ran  { A }  =/=  (/) )
3029biimpri 206 . . . . . . 7  |-  ( ran 
{ A }  =/=  (/) 
->  A  e.  ( _V  X.  _V ) )
3130necon1bi 2695 . . . . . 6  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ran  { A }  =  (/) )
3231unieqd 4250 . . . . 5  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  U. (/) )
33 uni0 4267 . . . . 5  |-  U. (/)  =  (/)
3432, 33syl6eq 2519 . . . 4  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  U. ran  { A }  =  (/) )
3528, 34eqtr4d 2506 . . 3  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  = 
U. ran  { A } )
36 2ndval 6779 . . 3  |-  ( 2nd `  A )  =  U. ran  { A }
3735, 36syl6eqr 2521 . 2  |-  ( -.  A  e.  ( _V 
X.  _V )  ->  ( { <. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
) )
3816, 37pm2.61i 164 1  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y } `
 A )  =  ( 2nd `  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   _Vcvv 3108   (/)c0 3780   {csn 4022   <.cop 4028   U.cuni 4240   {copab 4499    X. cxp 4992   dom cdm 4994   ran crn 4995   ` cfv 5581  (class class class)co 6277   {coprab 6278    |-> cmpt2 6279   2ndc2nd 6775
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-2nd 6777
This theorem is referenced by: (None)
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