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Theorem 1stval2 5813
Description: Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
Assertion
Ref Expression
1stval2  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )

Proof of Theorem 1stval2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 4428 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2605 . . . . . 6  |-  x  e. 
_V
3 vex 2605 . . . . . 6  |-  y  e. 
_V
42, 3op1st 5804 . . . . 5  |-  ( 1st `  <. x ,  y
>. )  =  x
52, 3op1stb 4235 . . . . 5  |-  |^| |^| <. x ,  y >.  =  x
64, 5eqtr4i 2105 . . . 4  |-  ( 1st `  <. x ,  y
>. )  =  |^| |^|
<. x ,  y >.
7 fveq2 5209 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  ( 1st `  <. x ,  y
>. ) )
8 inteq 3647 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| A  =  |^| <.
x ,  y >.
)
98inteqd 3649 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| A  =  |^| |^|
<. x ,  y >.
)
106, 7, 93eqtr4a 2140 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
1110exlimivv 1818 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
121, 11sylbi 119 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602   <.cop 3409   |^|cint 3644    X. cxp 4369   ` cfv 4932   1stc1st 5796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940  df-1st 5798
This theorem is referenced by:  1stdm  5839
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