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Theorem 2ndval2 6365
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
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4936 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2959 . . . . . 6  |-  x  e. 
_V
3 vex 2959 . . . . . 6  |-  y  e. 
_V
42, 3op2nd 6356 . . . . 5  |-  ( 2nd `  <. x ,  y
>. )  =  y
52, 3op2ndb 5353 . . . . 5  |-  |^| |^| |^| `' { <. x ,  y
>. }  =  y
64, 5eqtr4i 2459 . . . 4  |-  ( 2nd `  <. x ,  y
>. )  =  |^| |^|
|^| `' { <. x ,  y
>. }
7 fveq2 5728 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  ( 2nd `  <. x ,  y
>. ) )
8 sneq 3825 . . . . . . . 8  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
98cnveqd 5048 . . . . . . 7  |-  ( A  =  <. x ,  y
>.  ->  `' { A }  =  `' { <. x ,  y >. } )
109inteqd 4055 . . . . . 6  |-  ( A  =  <. x ,  y
>.  ->  |^| `' { A }  =  |^| `' { <. x ,  y >. } )
1110inteqd 4055 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| `' { A }  =  |^| |^| `' { <. x ,  y
>. } )
1211inteqd 4055 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| |^| `' { A }  =  |^| |^| |^| `' { <. x ,  y
>. } )
136, 7, 123eqtr4a 2494 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
1413exlimivv 1645 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
151, 14sylbi 188 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^|
|^| `' { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   <.cop 3817   |^|cint 4050    X. cxp 4876   `'ccnv 4877   ` cfv 5454   2ndc2nd 6348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-2nd 6350
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