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Theorem ucnimalem 20518
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 3116 . . . . . 6  |-  x  e. 
_V
3 vex 3116 . . . . . 6  |-  y  e. 
_V
42, 3op1std 6791 . . . . 5  |-  ( p  =  <. x ,  y
>.  ->  ( 1st `  p
)  =  x )
54fveq2d 5868 . . . 4  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 1st `  p ) )  =  ( F `
 x ) )
62, 3op2ndd 6792 . . . . 5  |-  ( p  =  <. x ,  y
>.  ->  ( 2nd `  p
)  =  y )
76fveq2d 5868 . . . 4  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 2nd `  p ) )  =  ( F `
 y ) )
85, 7opeq12d 4221 . . 3  |-  ( p  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >.  =  <. ( F `  x ) ,  ( F `  y ) >. )
98mpt2mpt 6376 . 2  |-  ( p  e.  ( X  X.  X )  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) )
>. )  =  (
x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)
101, 9eqtr4i 2499 1  |-  G  =  ( p  e.  ( X  X.  X ) 
|->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   <.cop 4033    |-> cmpt 4505    X. cxp 4997   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780  UnifOncust 20437   Cnucucn 20513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782
This theorem is referenced by:  ucnima  20519
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