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Theorem fpar 7281
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 7279 . . 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 7280 . . 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 3816 . 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 4932 . . . 4 |- <. ( F ` x ) , ( G ` y ) >. e. _V
65dfmpt2 7267 . . 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 5254 . . . . . . . 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 5254 . . . . . . . . . 10 |- ( ( { x } X. _V ) i^i ( _V X. { y } ) ) = ( ( { x } i^i _V ) X. ( _V i^i { y } ) )
9 inv1 3970 . . . . . . . . . . 11 |- ( { x } i^i _V ) = { x }
10 incom 3805 . . . . . . . . . . . 12 |- ( _V i^i { y } ) = ( { y } i^i _V )
11 inv1 3970 . . . . . . . . . . . 12 |- ( { y } i^i _V ) = { y }
1210, 11eqtri 2644 . . . . . . . . . . 11 |- ( _V i^i { y } ) = { y }
139, 12xpeq12i 5137 . . . . . . . . . 10 |- ( ( { x } i^i _V ) X. ( _V i^i { y } ) ) = ( { x } X. { y } )
14 vex 3203 . . . . . . . . . . 11 |- x e. _V
15 vex 3203 . . . . . . . . . . 11 |- y e. _V
1614, 15xpsn 6407 . . . . . . . . . 10 |- ( { x } X. { y } ) = { <. x , y >. }
178, 13, 163eqtri 2648 . . . . . . . . 9 |- ( ( { x } X. _V ) i^i ( _V X. { y } ) ) = { <. x , y >. }
18 inxp 5254 . . . . . . . . . 10 |- ( ( { ( F ` x ) } X. _V ) i^i ( _V X. { ( G ` y ) } ) ) = ( ( { ( F ` x ) } i^i _V ) X. ( _V i^i { ( G ` y ) } ) )
19 inv1 3970 . . . . . . . . . . 11 |- ( { ( F ` x ) } i^i _V ) = { ( F ` x ) }
20 incom 3805 . . . . . . . . . . . 12 |- ( _V i^i { ( G ` y ) } ) = ( { ( G ` y ) } i^i _V )
21 inv1 3970 . . . . . . . . . . . 12 |- ( { ( G ` y ) } i^i _V ) = { ( G ` y ) }
2220, 21eqtri 2644 . . . . . . . . . . 11 |- ( _V i^i { ( G ` y ) } ) = { ( G ` y ) }
2319, 22xpeq12i 5137 . . . . . . . . . 10 |- ( ( { ( F ` x ) } i^i _V ) X. ( _V i^i { ( G ` y ) } ) ) = ( { ( F ` x ) } X. { ( G ` y ) } )
24 fvex 6201 . . . . . . . . . . 11 |- ( F ` x ) e. _V
25 fvex 6201 . . . . . . . . . . 11 |- ( G ` y ) e. _V
2624, 25xpsn 6407 . . . . . . . . . 10 |- ( { ( F ` x ) } X. { ( G ` y ) } ) = { <. ( F ` x ) , ( G ` y ) >. }
2718, 23, 263eqtri 2648 . . . . . . . . 9 |- ( ( { ( F ` x ) } X. _V ) i^i ( _V X. { ( G ` y ) } ) ) = { <. ( F ` x ) , ( G ` y ) >. }
2817, 27xpeq12i 5137 . . . . . . . 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 4932 . . . . . . . . 9 |- <. x , y >. e. _V
3029, 5xpsn 6407 . . . . . . . 8 |- ( { <. x , y >. } X. { <. ( F ` x ) , ( G ` y ) >. } ) = { <. <. x , y >. , <. ( F ` x ) , ( G ` y ) >. >. }
317, 28, 303eqtri 2648 . . . . . . 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 11 . . . . . 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 4539 . . . . 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 11 . . . 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 4539 . . 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 4588 . . 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 2650 . 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 2681 1 |- ( ( F Fn A /\ G Fn B ) -> H = ( x e. A , y e. B |-> <. ( F ` x ) , ( G ` y ) >. ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   {csn 4177   <.cop 4183   U_ciun 4520   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   Fn wfn 5883   ` cfv 5888   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169
This theorem is referenced by: (None)
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