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Theorem brapply 25783
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 4405 . . . 4  |-  { ( A " { B } ) }  e.  _V
21inex1 4344 . . 3  |-  ( { ( A " { B } ) }  i^i  Singletons )  e.  _V
3 unieq 4024 . . . . 5  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  U. x  =  U. ( { ( A " { B } ) }  i^i  Singletons ) )
43unieqd 4026 . . . 4  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  U. U. x  = 
U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
54eqeq2d 2447 . . 3  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  ( C  = 
U. U. x  <->  C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) ) )
62, 5ceqsexv 2991 . 2  |-  ( E. x ( x  =  ( { ( A
" { B }
) }  i^i  Singletons )  /\  C  =  U. U. x )  <-> 
C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
7 df-apply 25717 . . . 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 4218 . . 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 4427 . . . 4  |-  <. A ,  B >.  e.  _V
10 brapply.3 . . . 4  |-  C  e. 
_V
119, 10brco 5043 . . 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 2959 . . . . . . 7  |-  x  e. 
_V
139, 12brco 5043 . . . . . 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 2959 . . . . . . . . . 10  |-  y  e. 
_V
159, 14brco 5043 . . . . . . . . 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 2959 . . . . . . . . . . . . 13  |-  z  e. 
_V
1916, 17, 18brpprod3a 25731 . . . . . . . . . . . 12  |-  ( <. A ,  B >.pprod (  _I  , Singleton ) z  <->  E. a E. b ( z  = 
<. a ,  b >.  /\  A  _I  a  /\  BSingleton b ) )
20 3anrot 941 . . . . . . . . . . . . . 14  |-  ( ( z  =  <. a ,  b >.  /\  A  _I  a  /\  BSingleton b
)  <->  ( A  _I  a  /\  BSingleton b  /\  z  =  <. a ,  b
>. ) )
21 vex 2959 . . . . . . . . . . . . . . . . 17  |-  a  e. 
_V
2221ideq 5025 . . . . . . . . . . . . . . . 16  |-  ( A  _I  a  <->  A  =  a )
23 eqcom 2438 . . . . . . . . . . . . . . . 16  |-  ( A  =  a  <->  a  =  A )
2422, 23bitri 241 . . . . . . . . . . . . . . 15  |-  ( A  _I  a  <->  a  =  A )
25 vex 2959 . . . . . . . . . . . . . . . 16  |-  b  e. 
_V
2617, 25brsingle 25762 . . . . . . . . . . . . . . 15  |-  ( BSingleton
b  <->  b  =  { B } )
27 biid 228 . . . . . . . . . . . . . . 15  |-  ( z  =  <. a ,  b
>. 
<->  z  =  <. a ,  b >. )
2824, 26, 273anbi123i 1142 . . . . . . . . . . . . . 14  |-  ( ( A  _I  a  /\  BSingleton b  /\  z  = 
<. a ,  b >.
)  <->  ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. ) )
2920, 28bitri 241 . . . . . . . . . . . . 13  |-  ( ( z  =  <. a ,  b >.  /\  A  _I  a  /\  BSingleton b
)  <->  ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. ) )
30292exbii 1593 . . . . . . . . . . . 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 4405 . . . . . . . . . . . . 13  |-  { B }  e.  _V
32 opeq1 3984 . . . . . . . . . . . . . 14  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
3332eqeq2d 2447 . . . . . . . . . . . . 13  |-  ( a  =  A  ->  (
z  =  <. a ,  b >.  <->  z  =  <. A ,  b >.
) )
34 opeq2 3985 . . . . . . . . . . . . . 14  |-  ( b  =  { B }  -> 
<. A ,  b >.  =  <. A ,  { B } >. )
3534eqeq2d 2447 . . . . . . . . . . . . 13  |-  ( b  =  { B }  ->  ( z  =  <. A ,  b >.  <->  z  =  <. A ,  { B } >. ) )
3616, 31, 33, 35ceqsex2v 2993 . . . . . . . . . . . 12  |-  ( E. a E. b ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. )  <->  z  =  <. A ,  { B } >. )
3719, 30, 363bitri 263 . . . . . . . . . . 11  |-  ( <. A ,  B >.pprod (  _I  , Singleton ) z  <->  z  =  <. A ,  { B } >. )
3837anbi1i 677 . . . . . . . . . 10  |-  ( (
<. A ,  B >.pprod (  _I  , Singleton ) z  /\  z (Singleton  o. Img ) y
)  <->  ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y ) )
3938exbii 1592 . . . . . . . . 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 4427 . . . . . . . . . . 11  |-  <. A ,  { B } >.  e.  _V
41 breq1 4215 . . . . . . . . . . 11  |-  ( z  =  <. A ,  { B } >.  ->  ( z (Singleton  o. Img ) y  <->  <. A ,  { B } >. (Singleton  o. Img ) y ) )
4240, 41ceqsexv 2991 . . . . . . . . . 10  |-  ( E. z ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y )  <->  <. A ,  { B } >. (Singleton  o. Img ) y )
4340, 14brco 5043 . . . . . . . . . 10  |-  ( <. A ,  { B } >. (Singleton  o. Img ) y  <->  E. x ( <. A ,  { B } >.Img x  /\  xSingleton y ) )
4416, 31, 12brimg 25782 . . . . . . . . . . . . 13  |-  ( <. A ,  { B } >.Img x  <->  x  =  ( A " { B } ) )
4512, 14brsingle 25762 . . . . . . . . . . . . 13  |-  ( xSingleton
y  <->  y  =  {
x } )
4644, 45anbi12i 679 . . . . . . . . . . . 12  |-  ( (
<. A ,  { B } >.Img x  /\  xSingleton y )  <->  ( x  =  ( A " { B } )  /\  y  =  { x } ) )
4746exbii 1592 . . . . . . . . . . 11  |-  ( E. x ( <. A ,  { B } >.Img x  /\  xSingleton y )  <->  E. x
( x  =  ( A " { B } )  /\  y  =  { x } ) )
48 imaexg 5217 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  ( A " { B }
)  e.  _V )
4916, 48ax-mp 8 . . . . . . . . . . . 12  |-  ( A
" { B }
)  e.  _V
50 sneq 3825 . . . . . . . . . . . . 13  |-  ( x  =  ( A " { B } )  ->  { x }  =  { ( A " { B } ) } )
5150eqeq2d 2447 . . . . . . . . . . . 12  |-  ( x  =  ( A " { B } )  -> 
( y  =  {
x }  <->  y  =  { ( A " { B } ) } ) )
5249, 51ceqsexv 2991 . . . . . . . . . . 11  |-  ( E. x ( x  =  ( A " { B } )  /\  y  =  { x } )  <-> 
y  =  { ( A " { B } ) } )
5347, 52bitri 241 . . . . . . . . . 10  |-  ( E. x ( <. A ,  { B } >.Img x  /\  xSingleton y )  <->  y  =  { ( A " { B } ) } )
5442, 43, 533bitri 263 . . . . . . . . 9  |-  ( E. z ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y )  <->  y  =  { ( A " { B } ) } )
5515, 39, 543bitri 263 . . . . . . . 8  |-  ( <. A ,  B >. ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) y  <-> 
y  =  { ( A " { B } ) } )
56 eqid 2436 . . . . . . . . 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 4909 . . . . . . . . . 10  |-  ( y ( _V  X.  _V ) x  <->  ( y  e. 
_V  /\  x  e.  _V ) )
5814, 12, 57mpbir2an 887 . . . . . . . . 9  |-  y ( _V  X.  _V )
x
59 epel 4497 . . . . . . . . . . 11  |-  ( z  _E  y  <->  z  e.  y )
6059anbi1i 677 . . . . . . . . . 10  |-  ( ( z  _E  y  /\  z  e.  Singletons )  <->  ( z  e.  y  /\  z  e. 
Singletons ) )
6114brres 5152 . . . . . . . . . 10  |-  ( z (  _E  |`  Singletons ) y  <->  ( z  _E  y  /\  z  e. 
Singletons ) )
62 elin 3530 . . . . . . . . . 10  |-  ( z  e.  ( y  i^i  Singletons
)  <->  ( z  e.  y  /\  z  e.  Singletons
) )
6360, 61, 623bitr4ri 270 . . . . . . . . 9  |-  ( z  e.  ( y  i^i  Singletons
)  <->  z (  _E  |` 
Singletons ) y )
6414, 12, 56, 58, 63brtxpsd3 25741 . . . . . . . 8  |-  ( y ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  |`  Singletons )  (x)  _V ) ) ) x  <-> 
x  =  ( y  i^i  Singletons ) )
6555, 64anbi12i 679 . . . . . . 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 1592 . . . . . 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 3535 . . . . . . . 8  |-  ( y  =  { ( A
" { B }
) }  ->  (
y  i^i  Singletons )  =  ( { ( A " { B } ) }  i^i  Singletons ) )
6867eqeq2d 2447 . . . . . . 7  |-  ( y  =  { ( A
" { B }
) }  ->  (
x  =  ( y  i^i  Singletons )  <->  x  =  ( { ( A " { B } ) }  i^i  Singletons ) ) )
691, 68ceqsexv 2991 . . . . . 6  |-  ( E. y ( y  =  { ( A " { B } ) }  /\  x  =  ( y  i^i  Singletons ) )  <->  x  =  ( { ( A " { B } ) }  i^i  Singletons ) )
7013, 66, 693bitri 263 . . . . 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 5043 . . . . . 6  |-  ( x ( Bigcup  o.  Bigcup ) C  <->  E. y ( x Bigcup y  /\  y Bigcup C ) )
7214brbigcup 25743 . . . . . . . . 9  |-  ( x
Bigcup y  <->  U. x  =  y )
73 eqcom 2438 . . . . . . . . 9  |-  ( U. x  =  y  <->  y  =  U. x )
7472, 73bitri 241 . . . . . . . 8  |-  ( x
Bigcup y  <->  y  =  U. x )
7510brbigcup 25743 . . . . . . . . 9  |-  ( y
Bigcup C  <->  U. y  =  C )
76 eqcom 2438 . . . . . . . . 9  |-  ( U. y  =  C  <->  C  =  U. y )
7775, 76bitri 241 . . . . . . . 8  |-  ( y
Bigcup C  <->  C  =  U. y )
7874, 77anbi12i 679 . . . . . . 7  |-  ( ( x Bigcup y  /\  y Bigcup C )  <->  ( y  =  U. x  /\  C  =  U. y ) )
7978exbii 1592 . . . . . 6  |-  ( E. y ( x Bigcup y  /\  y Bigcup C )  <->  E. y ( y  = 
U. x  /\  C  =  U. y ) )
8012uniex 4705 . . . . . . 7  |-  U. x  e.  _V
81 unieq 4024 . . . . . . . 8  |-  ( y  =  U. x  ->  U. y  =  U. U. x )
8281eqeq2d 2447 . . . . . . 7  |-  ( y  =  U. x  -> 
( C  =  U. y 
<->  C  =  U. U. x ) )
8380, 82ceqsexv 2991 . . . . . 6  |-  ( E. y ( y  = 
U. x  /\  C  =  U. y )  <->  C  =  U. U. x )
8471, 79, 833bitri 263 . . . . 5  |-  ( x ( Bigcup  o.  Bigcup ) C  <-> 
C  =  U. U. x )
8570, 84anbi12i 679 . . . 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 1592 . . 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 263 . 2  |-  ( <. A ,  B >.Apply C  <->  E. x ( x  =  ( { ( A
" { B }
) }  i^i  Singletons )  /\  C  =  U. U. x ) )
88 dffv5 25769 . . 3  |-  ( A `
 B )  = 
U. U. ( { ( A " { B } ) }  i^i  Singletons )
8988eqeq2i 2446 . 2  |-  ( C  =  ( A `  B )  <->  C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
906, 87, 893bitr4i 269 1  |-  ( <. A ,  B >.Apply C  <-> 
C  =  ( A `
 B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    i^i cin 3319   {csn 3814   <.cop 3817   U.cuni 4015   class class class wbr 4212    _E cep 4492    _I cid 4493    X. cxp 4876   ran crn 4879    |` cres 4880   "cima 4881    o. ccom 4882   ` cfv 5454  (++)csymdif 25662    (x) ctxp 25674  pprodcpprod 25675   Bigcupcbigcup 25678  Singletoncsingle 25682   Singletonscsingles 25683  Imgcimg 25686  Applycapply 25689
This theorem is referenced by:  dfrdg4  25795  tfrqfree  25796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-eprel 4494  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-symdif 25663  df-txp 25698  df-pprod 25699  df-bigcup 25702  df-singleton 25706  df-singles 25707  df-image 25708  df-cart 25709  df-img 25710  df-apply 25717
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