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Theorem ucnimalem 21077
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 3064 . . . . . 6  |-  x  e. 
_V
3 vex 3064 . . . . . 6  |-  y  e. 
_V
42, 3op1std 6796 . . . . 5  |-  ( p  =  <. x ,  y
>.  ->  ( 1st `  p
)  =  x )
54fveq2d 5855 . . . 4  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 1st `  p ) )  =  ( F `
 x ) )
62, 3op2ndd 6797 . . . . 5  |-  ( p  =  <. x ,  y
>.  ->  ( 2nd `  p
)  =  y )
76fveq2d 5855 . . . 4  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 2nd `  p ) )  =  ( F `
 y ) )
85, 7opeq12d 4169 . . 3  |-  ( p  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >.  =  <. ( F `  x ) ,  ( F `  y ) >. )
98mpt2mpt 6377 . 2  |-  ( p  e.  ( X  X.  X )  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) )
>. )  =  (
x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)
101, 9eqtr4i 2436 1  |-  G  =  ( p  e.  ( X  X.  X ) 
|->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844   <.cop 3980    |-> cmpt 4455    X. cxp 4823   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   1stc1st 6784   2ndc2nd 6785  UnifOncust 20996   Cnucucn 21072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-iota 5535  df-fun 5573  df-fv 5579  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787
This theorem is referenced by:  ucnima  21078
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