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Theorem 2ndval2 6037
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 4701 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2743 . . . . . 6  |-  x  e. 
_V
3 vex 2743 . . . . . 6  |-  y  e. 
_V
42, 3op2nd 6028 . . . . 5  |-  ( 2nd `  <. x ,  y
>. )  =  y
52, 3op2ndb 5108 . . . . 5  |-  |^| |^| |^| `' { <. x ,  y
>. }  =  y
64, 5eqtr4i 2279 . . . 4  |-  ( 2nd `  <. x ,  y
>. )  =  |^| |^|
|^| `' { <. x ,  y
>. }
7 fveq2 5423 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  ( 2nd `  <. x ,  y
>. ) )
8 sneq 3592 . . . . . . . 8  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
98cnveqd 4810 . . . . . . 7  |-  ( A  =  <. x ,  y
>.  ->  `' { A }  =  `' { <. x ,  y >. } )
109inteqd 3808 . . . . . 6  |-  ( A  =  <. x ,  y
>.  ->  |^| `' { A }  =  |^| `' { <. x ,  y >. } )
1110inteqd 3808 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| `' { A }  =  |^| |^| `' { <. x ,  y
>. } )
1211inteqd 3808 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| |^| `' { A }  =  |^| |^| |^| `' { <. x ,  y
>. } )
136, 7, 123eqtr4a 2314 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
1413exlimivv 2026 . 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 1537    = wceq 1619    e. wcel 1621   _Vcvv 2740   {csn 3581   <.cop 3584   |^|cint 3803    X. cxp 4624   `'ccnv 4625   ` cfv 4638   2ndc2nd 6020
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-2nd 6022
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