MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stval2 Structured version   Unicode version

Theorem 1stval2 6364
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 4936 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
2 vex 2959 . . . . . 6  |-  x  e. 
_V
3 vex 2959 . . . . . 6  |-  y  e. 
_V
42, 3op1st 6355 . . . . 5  |-  ( 1st `  <. x ,  y
>. )  =  x
52, 3op1stb 4758 . . . . 5  |-  |^| |^| <. x ,  y >.  =  x
64, 5eqtr4i 2459 . . . 4  |-  ( 1st `  <. x ,  y
>. )  =  |^| |^|
<. x ,  y >.
7 fveq2 5728 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  ( 1st `  <. x ,  y
>. ) )
8 inteq 4053 . . . . 5  |-  ( A  =  <. x ,  y
>.  ->  |^| A  =  |^| <.
x ,  y >.
)
98inteqd 4055 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  |^| |^| A  =  |^| |^|
<. x ,  y >.
)
106, 7, 93eqtr4a 2494 . . 3  |-  ( A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
1110exlimivv 1645 . 2  |-  ( E. x E. y  A  =  <. x ,  y
>.  ->  ( 1st `  A
)  =  |^| |^| A
)
121, 11sylbi 188 1  |-  ( A  e.  ( _V  X.  _V )  ->  ( 1st `  A )  =  |^| |^| A )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   |^|cint 4050    X. cxp 4876   ` cfv 5454   1stc1st 6347
This theorem is referenced by:  1stdm  6394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-1st 6349
  Copyright terms: Public domain W3C validator