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

Theorem ucnimalem 19855
Description: Reformulate the  G function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Hypotheses
Ref Expression
ucnprima.1  |-  ( ph  ->  U  e.  (UnifOn `  X ) )
ucnprima.2  |-  ( ph  ->  V  e.  (UnifOn `  Y ) )
ucnprima.3  |-  ( ph  ->  F  e.  ( U Cnu V ) )
ucnprima.4  |-  ( ph  ->  W  e.  V )
ucnprima.5  |-  G  =  ( x  e.  X ,  y  e.  X  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)
Assertion
Ref Expression
ucnimalem  |-  G  =  ( p  e.  ( X  X.  X ) 
|->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >. )
Distinct variable groups:    x, p, y, F    X, p, x, y
Allowed substitution hints:    ph( x, y, p)    U( x, y, p)    G( x, y, p)    V( x, y, p)    W( x, y, p)    Y( x, y, p)

Proof of Theorem ucnimalem
StepHypRef Expression
1 ucnprima.5 . 2  |-  G  =  ( x  e.  X ,  y  e.  X  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)
2 vex 2975 . . . . . 6  |-  x  e. 
_V
3 vex 2975 . . . . . 6  |-  y  e. 
_V
42, 3op1std 6587 . . . . 5  |-  ( p  =  <. x ,  y
>.  ->  ( 1st `  p
)  =  x )
54fveq2d 5695 . . . 4  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 1st `  p ) )  =  ( F `
 x ) )
62, 3op2ndd 6588 . . . . 5  |-  ( p  =  <. x ,  y
>.  ->  ( 2nd `  p
)  =  y )
76fveq2d 5695 . . . 4  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 2nd `  p ) )  =  ( F `
 y ) )
85, 7opeq12d 4067 . . 3  |-  ( p  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >.  =  <. ( F `  x ) ,  ( F `  y ) >. )
98mpt2mpt 6182 . 2  |-  ( p  e.  ( X  X.  X )  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) )
>. )  =  (
x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)
101, 9eqtr4i 2466 1  |-  G  =  ( p  e.  ( X  X.  X ) 
|->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   <.cop 3883    e. cmpt 4350    X. cxp 4838   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576  UnifOncust 19774   Cnucucn 19850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fv 5426  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578
This theorem is referenced by:  ucnima  19856
  Copyright terms: Public domain W3C validator