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Theorem 1stval2 5989
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
StepHypRef Expression
1 elvv 4655 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2730 . . . . . 6  |-  x  e. 
_V
3 vex 2730 . . . . . 6  |-  y  e. 
_V
42, 3op1st 5980 . . . . 5  |-  ( 1st `  <. x ,  y
>. )  =  x
52, 3op1stb 4460 . . . . 5  |-  |^| |^| <. x ,  y >.  =  x
64, 5eqtr4i 2276 . . . 4  |-  ( 1st `  <. x ,  y
>. )  =  |^| |^|
<. x ,  y >.
7 fveq2 5377 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  ( 1st `  <. x ,  y
>. ) )
8 inteq 3763 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| A  =  |^| <.
x ,  y >.
)
98inteqd 3765 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| A  =  |^| |^|
<. x ,  y >.
)
106, 7, 93eqtr4a 2311 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
1110exlimivv 2025 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
121, 11sylbi 189 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 6   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2727   <.cop 3547   |^|cint 3760    X. cxp 4578   ` cfv 4592   1stc1st 5972
This theorem is referenced by:  1stdm  6019
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-1st 5974
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