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Theorem 2ndval2 5783
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 4402 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2560 . . . . . 6  |-  x  e. 
_V
3 vex 2560 . . . . . 6  |-  y  e. 
_V
42, 3op2nd 5774 . . . . 5  |-  ( 2nd `  <. x ,  y
>. )  =  y
52, 3op2ndb 4804 . . . . 5  |-  |^| |^| |^| `' { <. x ,  y
>. }  =  y
64, 5eqtr4i 2063 . . . 4  |-  ( 2nd `  <. x ,  y
>. )  =  |^| |^|
|^| `' { <. x ,  y
>. }
7 fveq2 5178 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  ( 2nd `  <. x ,  y
>. ) )
8 sneq 3386 . . . . . . . 8  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
98cnveqd 4511 . . . . . . 7  |-  ( A  =  <. x ,  y
>.  ->  `' { A }  =  `' { <. x ,  y >. } )
109inteqd 3620 . . . . . 6  |-  ( A  =  <. x ,  y
>.  ->  |^| `' { A }  =  |^| `' { <. x ,  y >. } )
1110inteqd 3620 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| `' { A }  =  |^| |^| `' { <. x ,  y
>. } )
1211inteqd 3620 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| |^| `' { A }  =  |^| |^| |^| `' { <. x ,  y
>. } )
136, 7, 123eqtr4a 2098 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
1413exlimivv 1776 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 2nd `  A
)  =  |^| |^| |^| `' { A } )
151, 14sylbi 114 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 2nd `  A )  =  |^| |^|
|^| `' { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2557   {csn 3375   <.cop 3378   |^|cint 3615    X. cxp 4343   `'ccnv 4344   ` cfv 4902   2ndc2nd 5766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768
This theorem is referenced by: (None)
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