MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ndval2 Unicode version

Theorem 2ndval2 6099
Description: Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
Assertion
Ref Expression
2ndval2  |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^|
|^| `' { A } )

Proof of Theorem 2ndval2
StepHypRef Expression
1 elvv 4747 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2792 . . . . . 6  |-  x  e. 
_V
3 vex 2792 . . . . . 6  |-  y  e. 
_V
42, 3op2nd 6090 . . . . 5  |-  ( 2nd `  <. x ,  y
>. )  =  y
52, 3op2ndb 5154 . . . . 5  |-  |^| |^| |^| `' { <. x ,  y
>. }  =  y
64, 5eqtr4i 2307 . . . 4  |-  ( 2nd `  <. x ,  y
>. )  =  |^| |^|
|^| `' { <. x ,  y
>. }
7 fveq2 5485 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  ( 2nd `  <. x ,  y
>. ) )
8 sneq 3652 . . . . . . . 8  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
98cnveqd 4856 . . . . . . 7  |-  ( A  =  <. x ,  y
>.  ->  `' { A }  =  `' { <. x ,  y >. } )
109inteqd 3868 . . . . . 6  |-  ( A  =  <. x ,  y
>.  ->  |^| `' { A }  =  |^| `' { <. x ,  y >. } )
1110inteqd 3868 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| `' { A }  =  |^| |^| `' { <. x ,  y
>. } )
1211inteqd 3868 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| |^| `' { A }  =  |^| |^| |^| `' { <. x ,  y
>. } )
136, 7, 123eqtr4a 2342 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
1413exlimivv 1671 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
151, 14sylbi 189 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^|
|^| `' { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 6   E.wex 1533    = wceq 1628    e. wcel 1688   _Vcvv 2789   {csn 3641   <.cop 3644   |^|cint 3863    X. cxp 4686   `'ccnv 4687   ` cfv 5221   2ndc2nd 6082
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229  df-2nd 6084
  Copyright terms: Public domain W3C validator