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Theorem brapply 30753
Description: The binary relationship form of the Apply function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brapply.1  |-  A  e. 
_V
brapply.2  |-  B  e. 
_V
brapply.3  |-  C  e. 
_V
Assertion
Ref Expression
brapply  |-  ( <. A ,  B >.Apply C  <-> 
C  =  ( A `
 B ) )

Proof of Theorem brapply
Dummy variables  a 
b  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4654 . . . 4  |-  { ( A " { B } ) }  e.  _V
21inex1 4557 . . 3  |-  ( { ( A " { B } ) }  i^i  Singletons )  e.  _V
3 unieq 4219 . . . . 5  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  U. x  =  U. ( { ( A " { B } ) }  i^i  Singletons ) )
43unieqd 4221 . . . 4  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  U. U. x  = 
U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
54eqeq2d 2471 . . 3  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  ( C  = 
U. U. x  <->  C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) ) )
62, 5ceqsexv 3095 . 2  |-  ( E. x ( x  =  ( { ( A
" { B }
) }  i^i  Singletons )  /\  C  =  U. U. x )  <-> 
C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
7 df-apply 30687 . . . 4  |- Apply  =  ( ( Bigcup  o.  Bigcup )  o.  ( ( ( _V 
X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )  o.  (
(Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) )
87breqi 4421 . . 3  |-  ( <. A ,  B >.Apply C  <->  <. A ,  B >. ( ( Bigcup  o.  Bigcup )  o.  ( ( ( _V 
X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )  o.  (
(Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) ) C )
9 opex 4677 . . . 4  |-  <. A ,  B >.  e.  _V
10 brapply.3 . . . 4  |-  C  e. 
_V
119, 10brco 5023 . . 3  |-  ( <. A ,  B >. ( ( Bigcup  o.  Bigcup )  o.  ( ( ( _V 
X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )  o.  (
(Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) ) C  <->  E. x
( <. A ,  B >. ( ( ( _V 
X.  _V )  \  ran  ( ( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )  o.  (
(Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) x  /\  x (
Bigcup  o.  Bigcup ) C ) )
12 vex 3059 . . . . . . 7  |-  x  e. 
_V
139, 12brco 5023 . . . . . 6  |-  ( <. A ,  B >. ( ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )  o.  (
(Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) x  <->  E. y ( <. A ,  B >. ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) y  /\  y ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )  /_\  ( (  _E  |`  Singletons )  (x)  _V ) ) ) x ) )
14 vex 3059 . . . . . . . . . 10  |-  y  e. 
_V
159, 14brco 5023 . . . . . . . . 9  |-  ( <. A ,  B >. ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) y  <->  E. z ( <. A ,  B >.pprod (  _I  , Singleton ) z  /\  z (Singleton  o. Img ) y ) )
16 brapply.1 . . . . . . . . . . . . 13  |-  A  e. 
_V
17 brapply.2 . . . . . . . . . . . . 13  |-  B  e. 
_V
18 vex 3059 . . . . . . . . . . . . 13  |-  z  e. 
_V
1916, 17, 18brpprod3a 30701 . . . . . . . . . . . 12  |-  ( <. A ,  B >.pprod (  _I  , Singleton ) z  <->  E. a E. b ( z  = 
<. a ,  b >.  /\  A  _I  a  /\  BSingleton b ) )
20 3anrot 996 . . . . . . . . . . . . . 14  |-  ( ( z  =  <. a ,  b >.  /\  A  _I  a  /\  BSingleton b
)  <->  ( A  _I  a  /\  BSingleton b  /\  z  =  <. a ,  b
>. ) )
21 vex 3059 . . . . . . . . . . . . . . . . 17  |-  a  e. 
_V
2221ideq 5005 . . . . . . . . . . . . . . . 16  |-  ( A  _I  a  <->  A  =  a )
23 eqcom 2468 . . . . . . . . . . . . . . . 16  |-  ( A  =  a  <->  a  =  A )
2422, 23bitri 257 . . . . . . . . . . . . . . 15  |-  ( A  _I  a  <->  a  =  A )
25 vex 3059 . . . . . . . . . . . . . . . 16  |-  b  e. 
_V
2617, 25brsingle 30732 . . . . . . . . . . . . . . 15  |-  ( BSingleton
b  <->  b  =  { B } )
27 biid 244 . . . . . . . . . . . . . . 15  |-  ( z  =  <. a ,  b
>. 
<->  z  =  <. a ,  b >. )
2824, 26, 273anbi123i 1203 . . . . . . . . . . . . . 14  |-  ( ( A  _I  a  /\  BSingleton b  /\  z  = 
<. a ,  b >.
)  <->  ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. ) )
2920, 28bitri 257 . . . . . . . . . . . . 13  |-  ( ( z  =  <. a ,  b >.  /\  A  _I  a  /\  BSingleton b
)  <->  ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. ) )
30292exbii 1729 . . . . . . . . . . . 12  |-  ( E. a E. b ( z  =  <. a ,  b >.  /\  A  _I  a  /\  BSingleton b
)  <->  E. a E. b
( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b
>. ) )
31 snex 4654 . . . . . . . . . . . . 13  |-  { B }  e.  _V
32 opeq1 4179 . . . . . . . . . . . . . 14  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
3332eqeq2d 2471 . . . . . . . . . . . . 13  |-  ( a  =  A  ->  (
z  =  <. a ,  b >.  <->  z  =  <. A ,  b >.
) )
34 opeq2 4180 . . . . . . . . . . . . . 14  |-  ( b  =  { B }  -> 
<. A ,  b >.  =  <. A ,  { B } >. )
3534eqeq2d 2471 . . . . . . . . . . . . 13  |-  ( b  =  { B }  ->  ( z  =  <. A ,  b >.  <->  z  =  <. A ,  { B } >. ) )
3616, 31, 33, 35ceqsex2v 3098 . . . . . . . . . . . 12  |-  ( E. a E. b ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. )  <->  z  =  <. A ,  { B } >. )
3719, 30, 363bitri 279 . . . . . . . . . . 11  |-  ( <. A ,  B >.pprod (  _I  , Singleton ) z  <->  z  =  <. A ,  { B } >. )
3837anbi1i 706 . . . . . . . . . 10  |-  ( (
<. A ,  B >.pprod (  _I  , Singleton ) z  /\  z (Singleton  o. Img ) y
)  <->  ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y ) )
3938exbii 1728 . . . . . . . . 9  |-  ( E. z ( <. A ,  B >.pprod (  _I  , Singleton ) z  /\  z (Singleton  o. Img ) y )  <->  E. z
( z  =  <. A ,  { B } >.  /\  z (Singleton  o. Img ) y ) )
40 opex 4677 . . . . . . . . . . 11  |-  <. A ,  { B } >.  e.  _V
41 breq1 4418 . . . . . . . . . . 11  |-  ( z  =  <. A ,  { B } >.  ->  ( z (Singleton  o. Img ) y  <->  <. A ,  { B } >. (Singleton  o. Img ) y ) )
4240, 41ceqsexv 3095 . . . . . . . . . 10  |-  ( E. z ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y )  <->  <. A ,  { B } >. (Singleton  o. Img ) y )
4340, 14brco 5023 . . . . . . . . . 10  |-  ( <. A ,  { B } >. (Singleton  o. Img ) y  <->  E. x ( <. A ,  { B } >.Img x  /\  xSingleton y ) )
4416, 31, 12brimg 30752 . . . . . . . . . . . . 13  |-  ( <. A ,  { B } >.Img x  <->  x  =  ( A " { B } ) )
4512, 14brsingle 30732 . . . . . . . . . . . . 13  |-  ( xSingleton
y  <->  y  =  {
x } )
4644, 45anbi12i 708 . . . . . . . . . . . 12  |-  ( (
<. A ,  { B } >.Img x  /\  xSingleton y )  <->  ( x  =  ( A " { B } )  /\  y  =  { x } ) )
4746exbii 1728 . . . . . . . . . . 11  |-  ( E. x ( <. A ,  { B } >.Img x  /\  xSingleton y )  <->  E. x
( x  =  ( A " { B } )  /\  y  =  { x } ) )
48 imaexg 6756 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  ( A " { B }
)  e.  _V )
4916, 48ax-mp 5 . . . . . . . . . . . 12  |-  ( A
" { B }
)  e.  _V
50 sneq 3989 . . . . . . . . . . . . 13  |-  ( x  =  ( A " { B } )  ->  { x }  =  { ( A " { B } ) } )
5150eqeq2d 2471 . . . . . . . . . . . 12  |-  ( x  =  ( A " { B } )  -> 
( y  =  {
x }  <->  y  =  { ( A " { B } ) } ) )
5249, 51ceqsexv 3095 . . . . . . . . . . 11  |-  ( E. x ( x  =  ( A " { B } )  /\  y  =  { x } )  <-> 
y  =  { ( A " { B } ) } )
5347, 52bitri 257 . . . . . . . . . 10  |-  ( E. x ( <. A ,  { B } >.Img x  /\  xSingleton y )  <->  y  =  { ( A " { B } ) } )
5442, 43, 533bitri 279 . . . . . . . . 9  |-  ( E. z ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y )  <->  y  =  { ( A " { B } ) } )
5515, 39, 543bitri 279 . . . . . . . 8  |-  ( <. A ,  B >. ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) y  <-> 
y  =  { ( A " { B } ) } )
56 eqid 2461 . . . . . . . . 9  |-  ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )  /_\  ( (  _E  |`  Singletons )  (x)  _V ) ) )  =  ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )
57 brxp 4883 . . . . . . . . . 10  |-  ( y ( _V  X.  _V ) x  <->  ( y  e. 
_V  /\  x  e.  _V ) )
5814, 12, 57mpbir2an 936 . . . . . . . . 9  |-  y ( _V  X.  _V )
x
59 epel 4766 . . . . . . . . . . 11  |-  ( z  _E  y  <->  z  e.  y )
6059anbi1i 706 . . . . . . . . . 10  |-  ( ( z  _E  y  /\  z  e.  Singletons )  <->  ( z  e.  y  /\  z  e. 
Singletons ) )
6114brres 5129 . . . . . . . . . 10  |-  ( z (  _E  |`  Singletons ) y  <->  ( z  _E  y  /\  z  e. 
Singletons ) )
62 elin 3628 . . . . . . . . . 10  |-  ( z  e.  ( y  i^i  Singletons
)  <->  ( z  e.  y  /\  z  e.  Singletons
) )
6360, 61, 623bitr4ri 286 . . . . . . . . 9  |-  ( z  e.  ( y  i^i  Singletons
)  <->  z (  _E  |` 
Singletons ) y )
6414, 12, 56, 58, 63brtxpsd3 30711 . . . . . . . 8  |-  ( y ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) ) x  <->  x  =  ( y  i^i  Singletons ) )
6555, 64anbi12i 708 . . . . . . 7  |-  ( (
<. A ,  B >. ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) y  /\  y ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )  /_\  ( (  _E  |`  Singletons )  (x)  _V ) ) ) x )  <->  ( y  =  { ( A " { B } ) }  /\  x  =  ( y  i^i  Singletons ) ) )
6665exbii 1728 . . . . . 6  |-  ( E. y ( <. A ,  B >. ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) y  /\  y
( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) ) x )  <->  E. y ( y  =  { ( A " { B } ) }  /\  x  =  ( y  i^i  Singletons ) ) )
67 ineq1 3638 . . . . . . . 8  |-  ( y  =  { ( A
" { B }
) }  ->  (
y  i^i  Singletons )  =  ( { ( A " { B } ) }  i^i  Singletons ) )
6867eqeq2d 2471 . . . . . . 7  |-  ( y  =  { ( A
" { B }
) }  ->  (
x  =  ( y  i^i  Singletons )  <->  x  =  ( { ( A " { B } ) }  i^i  Singletons ) ) )
691, 68ceqsexv 3095 . . . . . 6  |-  ( E. y ( y  =  { ( A " { B } ) }  /\  x  =  ( y  i^i  Singletons ) )  <->  x  =  ( { ( A " { B } ) }  i^i  Singletons ) )
7013, 66, 693bitri 279 . . . . 5  |-  ( <. A ,  B >. ( ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )  o.  (
(Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) x  <->  x  =  ( { ( A " { B } ) }  i^i  Singletons ) )
7112, 10brco 5023 . . . . . 6  |-  ( x ( Bigcup  o.  Bigcup ) C  <->  E. y ( x Bigcup y  /\  y Bigcup C ) )
7214brbigcup 30713 . . . . . . . . 9  |-  ( x
Bigcup y  <->  U. x  =  y )
73 eqcom 2468 . . . . . . . . 9  |-  ( U. x  =  y  <->  y  =  U. x )
7472, 73bitri 257 . . . . . . . 8  |-  ( x
Bigcup y  <->  y  =  U. x )
7510brbigcup 30713 . . . . . . . . 9  |-  ( y
Bigcup C  <->  U. y  =  C )
76 eqcom 2468 . . . . . . . . 9  |-  ( U. y  =  C  <->  C  =  U. y )
7775, 76bitri 257 . . . . . . . 8  |-  ( y
Bigcup C  <->  C  =  U. y )
7874, 77anbi12i 708 . . . . . . 7  |-  ( ( x Bigcup y  /\  y Bigcup C )  <->  ( y  =  U. x  /\  C  =  U. y ) )
7978exbii 1728 . . . . . 6  |-  ( E. y ( x Bigcup y  /\  y Bigcup C )  <->  E. y ( y  = 
U. x  /\  C  =  U. y ) )
8012uniex 6613 . . . . . . 7  |-  U. x  e.  _V
81 unieq 4219 . . . . . . . 8  |-  ( y  =  U. x  ->  U. y  =  U. U. x )
8281eqeq2d 2471 . . . . . . 7  |-  ( y  =  U. x  -> 
( C  =  U. y 
<->  C  =  U. U. x ) )
8380, 82ceqsexv 3095 . . . . . 6  |-  ( E. y ( y  = 
U. x  /\  C  =  U. y )  <->  C  =  U. U. x )
8471, 79, 833bitri 279 . . . . 5  |-  ( x ( Bigcup  o.  Bigcup ) C  <-> 
C  =  U. U. x )
8570, 84anbi12i 708 . . . 4  |-  ( (
<. A ,  B >. ( ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )  /_\  ( (  _E  |` 
Singletons )  (x)  _V )
) )  o.  (
(Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) x  /\  x (
Bigcup  o.  Bigcup ) C )  <-> 
( x  =  ( { ( A " { B } ) }  i^i  Singletons )  /\  C  =  U. U. x ) )
8685exbii 1728 . . 3  |-  ( E. x ( <. A ,  B >. ( ( ( _V  X.  _V )  \  ran  ( ( _V 
(x)  _E  )  /_\  ( (  _E  |`  Singletons )  (x)  _V ) ) )  o.  ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) ) x  /\  x
( Bigcup  o.  Bigcup ) C )  <->  E. x ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  /\  C  =  U. U. x ) )
878, 11, 863bitri 279 . 2  |-  ( <. A ,  B >.Apply C  <->  E. x ( x  =  ( { ( A
" { B }
) }  i^i  Singletons )  /\  C  =  U. U. x ) )
88 dffv5 30739 . . 3  |-  ( A `
 B )  = 
U. U. ( { ( A " { B } ) }  i^i  Singletons )
8988eqeq2i 2473 . 2  |-  ( C  =  ( A `  B )  <->  C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
906, 87, 893bitr4i 285 1  |-  ( <. A ,  B >.Apply C  <-> 
C  =  ( A `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454   E.wex 1673    e. wcel 1897   _Vcvv 3056    \ cdif 3412    i^i cin 3414    /_\ csymdif 3673   {csn 3979   <.cop 3985   U.cuni 4211   class class class wbr 4415    _E cep 4761    _I cid 4762    X. cxp 4850   ran crn 4853    |` cres 4854   "cima 4855    o. ccom 4856   ` cfv 5600    (x) ctxp 30644  pprodcpprod 30645   Bigcupcbigcup 30648  Singletoncsingle 30652   Singletonscsingles 30653  Imgcimg 30656  Applycapply 30659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-symdif 3674  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-eprel 4763  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-fo 5606  df-fv 5608  df-1st 6819  df-2nd 6820  df-txp 30668  df-pprod 30669  df-bigcup 30672  df-singleton 30676  df-singles 30677  df-image 30678  df-cart 30679  df-img 30680  df-apply 30687
This theorem is referenced by:  dfrecs2  30765  dfrdg4  30766
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