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Theorem 1stval2 6790
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 5047 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 3109 . . . . . 6  |-  x  e. 
_V
3 vex 3109 . . . . . 6  |-  y  e. 
_V
42, 3op1st 6781 . . . . 5  |-  ( 1st `  <. x ,  y
>. )  =  x
52, 3op1stb 4707 . . . . 5  |-  |^| |^| <. x ,  y >.  =  x
64, 5eqtr4i 2486 . . . 4  |-  ( 1st `  <. x ,  y
>. )  =  |^| |^|
<. x ,  y >.
7 fveq2 5848 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  ( 1st `  <. x ,  y
>. ) )
8 inteq 4274 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| A  =  |^| <.
x ,  y >.
)
98inteqd 4276 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| A  =  |^| |^|
<. x ,  y >.
)
106, 7, 93eqtr4a 2521 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
1110exlimivv 1728 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
121, 11sylbi 195 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106   <.cop 4022   |^|cint 4271    X. cxp 4986   ` cfv 5570   1stc1st 6771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773
This theorem is referenced by:  1stdm  6820
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