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Theorem fpar 6238
Description: Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as  z  =  ( ( sqr `  x
)  +  ( abs `  y ) ). (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
Hypothesis
Ref Expression
fpar.1  |-  H  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
Assertion
Ref Expression
fpar  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  H  =  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )
)
Distinct variable groups:    x, y, A    x, B, y    x, F, y    x, G, y
Allowed substitution hints:    H( x, y)

Proof of Theorem fpar
StepHypRef Expression
1 fparlem3 6236 . . 3  |-  ( F  Fn  A  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  =  U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) ) )
2 fparlem4 6237 . . 3  |-  ( G  Fn  B  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  U_ y  e.  B  ( ( _V  X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) ) )
31, 2ineqan12d 3385 . 2  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )  =  ( U_ x  e.  A  (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  U_ y  e.  B  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) ) )
4 fpar.1 . 2  |-  H  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) ) )
5 opex 4253 . . . 4  |-  <. ( F `  x ) ,  ( G `  y ) >.  e.  _V
65dfmpt2 6225 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y
>. ,  <. ( F `
 x ) ,  ( G `  y
) >. >. }
7 inxp 4834 . . . . . . . 8  |-  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  ( ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  X.  (
( { ( F `
 x ) }  X.  _V )  i^i  ( _V  X.  {
( G `  y
) } ) ) )
8 inxp 4834 . . . . . . . . . 10  |-  ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  =  ( ( { x }  i^i  _V )  X.  ( _V  i^i  { y } ) )
9 inv1 3494 . . . . . . . . . . 11  |-  ( { x }  i^i  _V )  =  { x }
10 incom 3374 . . . . . . . . . . . 12  |-  ( _V 
i^i  { y } )  =  ( { y }  i^i  _V )
11 inv1 3494 . . . . . . . . . . . 12  |-  ( { y }  i^i  _V )  =  { y }
1210, 11eqtri 2316 . . . . . . . . . . 11  |-  ( _V 
i^i  { y } )  =  { y }
139, 12xpeq12i 4727 . . . . . . . . . 10  |-  ( ( { x }  i^i  _V )  X.  ( _V 
i^i  { y } ) )  =  ( { x }  X.  {
y } )
14 vex 2804 . . . . . . . . . . 11  |-  x  e. 
_V
15 vex 2804 . . . . . . . . . . 11  |-  y  e. 
_V
1614, 15xpsn 5716 . . . . . . . . . 10  |-  ( { x }  X.  {
y } )  =  { <. x ,  y
>. }
178, 13, 163eqtri 2320 . . . . . . . . 9  |-  ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  =  { <. x ,  y >. }
18 inxp 4834 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  X.  _V )  i^i  ( _V  X.  { ( G `
 y ) } ) )  =  ( ( { ( F `
 x ) }  i^i  _V )  X.  ( _V  i^i  {
( G `  y
) } ) )
19 inv1 3494 . . . . . . . . . . 11  |-  ( { ( F `  x
) }  i^i  _V )  =  { ( F `  x ) }
20 incom 3374 . . . . . . . . . . . 12  |-  ( _V 
i^i  { ( G `  y ) } )  =  ( { ( G `  y ) }  i^i  _V )
21 inv1 3494 . . . . . . . . . . . 12  |-  ( { ( G `  y
) }  i^i  _V )  =  { ( G `  y ) }
2220, 21eqtri 2316 . . . . . . . . . . 11  |-  ( _V 
i^i  { ( G `  y ) } )  =  { ( G `
 y ) }
2319, 22xpeq12i 4727 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  i^i  _V )  X.  ( _V 
i^i  { ( G `  y ) } ) )  =  ( { ( F `  x
) }  X.  {
( G `  y
) } )
24 fvex 5555 . . . . . . . . . . 11  |-  ( F `
 x )  e. 
_V
25 fvex 5555 . . . . . . . . . . 11  |-  ( G `
 y )  e. 
_V
2624, 25xpsn 5716 . . . . . . . . . 10  |-  ( { ( F `  x
) }  X.  {
( G `  y
) } )  =  { <. ( F `  x ) ,  ( G `  y )
>. }
2718, 23, 263eqtri 2320 . . . . . . . . 9  |-  ( ( { ( F `  x ) }  X.  _V )  i^i  ( _V  X.  { ( G `
 y ) } ) )  =  { <. ( F `  x
) ,  ( G `
 y ) >. }
2817, 27xpeq12i 4727 . . . . . . . 8  |-  ( ( ( { x }  X.  _V )  i^i  ( _V  X.  { y } ) )  X.  (
( { ( F `
 x ) }  X.  _V )  i^i  ( _V  X.  {
( G `  y
) } ) ) )  =  ( {
<. x ,  y >. }  X.  { <. ( F `  x ) ,  ( G `  y ) >. } )
29 opex 4253 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
3029, 5xpsn 5716 . . . . . . . 8  |-  ( {
<. x ,  y >. }  X.  { <. ( F `  x ) ,  ( G `  y ) >. } )  =  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. }
317, 28, 303eqtri 2320 . . . . . . 7  |-  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  { <. <.
x ,  y >. ,  <. ( F `  x ) ,  ( G `  y )
>. >. }
3231a1i 10 . . . . . 6  |-  ( y  e.  B  ->  (
( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )  i^i  ( ( _V 
X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) ) )  =  { <. <. x ,  y
>. ,  <. ( F `
 x ) ,  ( G `  y
) >. >. } )
3332iuneq2i 3939 . . . . 5  |-  U_ y  e.  B  ( (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  U_ y  e.  B  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. }
3433a1i 10 . . . 4  |-  ( x  e.  A  ->  U_ y  e.  B  ( (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  U_ y  e.  B  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. } )
3534iuneq2i 3939 . . 3  |-  U_ x  e.  A  U_ y  e.  B  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  U_ x  e.  A  U_ y  e.  B  { <. <. x ,  y >. ,  <. ( F `  x ) ,  ( G `  y ) >. >. }
36 2iunin 3986 . . 3  |-  U_ x  e.  A  U_ y  e.  B  ( ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )  =  ( U_ x  e.  A  (
( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  U_ y  e.  B  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )
376, 35, 363eqtr2i 2322 . 2  |-  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )  =  ( U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  i^i  U_ y  e.  B  (
( _V  X.  {
y } )  X.  ( _V  X.  {
( G `  y
) } ) ) )
383, 4, 373eqtr4g 2353 1  |-  ( ( F  Fn  A  /\  G  Fn  B )  ->  H  =  ( x  e.  A ,  y  e.  B  |->  <. ( F `  x ) ,  ( G `  y ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   {csn 3653   <.cop 3656   U_ciun 3921    X. cxp 4703   `'ccnv 4704    |` cres 4707    o. ccom 4709    Fn wfn 5266   ` cfv 5271    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139
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