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Theorem 2ndval2 6138
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 4748 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2791 . . . . . 6  |-  x  e. 
_V
3 vex 2791 . . . . . 6  |-  y  e. 
_V
42, 3op2nd 6129 . . . . 5  |-  ( 2nd `  <. x ,  y
>. )  =  y
52, 3op2ndb 5156 . . . . 5  |-  |^| |^| |^| `' { <. x ,  y
>. }  =  y
64, 5eqtr4i 2306 . . . 4  |-  ( 2nd `  <. x ,  y
>. )  =  |^| |^|
|^| `' { <. x ,  y
>. }
7 fveq2 5525 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  ( 2nd `  <. x ,  y
>. ) )
8 sneq 3651 . . . . . . . 8  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
98cnveqd 4857 . . . . . . 7  |-  ( A  =  <. x ,  y
>.  ->  `' { A }  =  `' { <. x ,  y >. } )
109inteqd 3867 . . . . . 6  |-  ( A  =  <. x ,  y
>.  ->  |^| `' { A }  =  |^| `' { <. x ,  y >. } )
1110inteqd 3867 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| `' { A }  =  |^| |^| `' { <. x ,  y
>. } )
1211inteqd 3867 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| |^| `' { A }  =  |^| |^| |^| `' { <. x ,  y
>. } )
136, 7, 123eqtr4a 2341 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
1413exlimivv 1667 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
151, 14sylbi 187 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^|
|^| `' { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643   |^|cint 3862    X. cxp 4687   `'ccnv 4688   ` cfv 5255   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-int 3863  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-2nd 6123
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