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Theorem brapply 29153
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 4683 . . . 4  |-  { ( A " { B } ) }  e.  _V
21inex1 4583 . . 3  |-  ( { ( A " { B } ) }  i^i  Singletons )  e.  _V
3 unieq 4248 . . . . 5  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  U. x  =  U. ( { ( A " { B } ) }  i^i  Singletons ) )
43unieqd 4250 . . . 4  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  U. U. x  = 
U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
54eqeq2d 2476 . . 3  |-  ( x  =  ( { ( A " { B } ) }  i^i  Singletons )  ->  ( C  = 
U. U. x  <->  C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) ) )
62, 5ceqsexv 3145 . 2  |-  ( E. x ( x  =  ( { ( A
" { B }
) }  i^i  Singletons )  /\  C  =  U. U. x )  <-> 
C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
7 df-apply 29087 . . . 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 4448 . . 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 4706 . . . 4  |-  <. A ,  B >.  e.  _V
10 brapply.3 . . . 4  |-  C  e. 
_V
119, 10brco 5166 . . 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 3111 . . . . . . 7  |-  x  e. 
_V
139, 12brco 5166 . . . . . 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 3111 . . . . . . . . . 10  |-  y  e. 
_V
159, 14brco 5166 . . . . . . . . 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 3111 . . . . . . . . . . . . 13  |-  z  e. 
_V
1916, 17, 18brpprod3a 29101 . . . . . . . . . . . 12  |-  ( <. A ,  B >.pprod (  _I  , Singleton ) z  <->  E. a E. b ( z  = 
<. a ,  b >.  /\  A  _I  a  /\  BSingleton b ) )
20 3anrot 973 . . . . . . . . . . . . . 14  |-  ( ( z  =  <. a ,  b >.  /\  A  _I  a  /\  BSingleton b
)  <->  ( A  _I  a  /\  BSingleton b  /\  z  =  <. a ,  b
>. ) )
21 vex 3111 . . . . . . . . . . . . . . . . 17  |-  a  e. 
_V
2221ideq 5148 . . . . . . . . . . . . . . . 16  |-  ( A  _I  a  <->  A  =  a )
23 eqcom 2471 . . . . . . . . . . . . . . . 16  |-  ( A  =  a  <->  a  =  A )
2422, 23bitri 249 . . . . . . . . . . . . . . 15  |-  ( A  _I  a  <->  a  =  A )
25 vex 3111 . . . . . . . . . . . . . . . 16  |-  b  e. 
_V
2617, 25brsingle 29132 . . . . . . . . . . . . . . 15  |-  ( BSingleton
b  <->  b  =  { B } )
27 biid 236 . . . . . . . . . . . . . . 15  |-  ( z  =  <. a ,  b
>. 
<->  z  =  <. a ,  b >. )
2824, 26, 273anbi123i 1180 . . . . . . . . . . . . . 14  |-  ( ( A  _I  a  /\  BSingleton b  /\  z  = 
<. a ,  b >.
)  <->  ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. ) )
2920, 28bitri 249 . . . . . . . . . . . . 13  |-  ( ( z  =  <. a ,  b >.  /\  A  _I  a  /\  BSingleton b
)  <->  ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. ) )
30292exbii 1640 . . . . . . . . . . . 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 4683 . . . . . . . . . . . . 13  |-  { B }  e.  _V
32 opeq1 4208 . . . . . . . . . . . . . 14  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
3332eqeq2d 2476 . . . . . . . . . . . . 13  |-  ( a  =  A  ->  (
z  =  <. a ,  b >.  <->  z  =  <. A ,  b >.
) )
34 opeq2 4209 . . . . . . . . . . . . . 14  |-  ( b  =  { B }  -> 
<. A ,  b >.  =  <. A ,  { B } >. )
3534eqeq2d 2476 . . . . . . . . . . . . 13  |-  ( b  =  { B }  ->  ( z  =  <. A ,  b >.  <->  z  =  <. A ,  { B } >. ) )
3616, 31, 33, 35ceqsex2v 3147 . . . . . . . . . . . 12  |-  ( E. a E. b ( a  =  A  /\  b  =  { B }  /\  z  =  <. a ,  b >. )  <->  z  =  <. A ,  { B } >. )
3719, 30, 363bitri 271 . . . . . . . . . . 11  |-  ( <. A ,  B >.pprod (  _I  , Singleton ) z  <->  z  =  <. A ,  { B } >. )
3837anbi1i 695 . . . . . . . . . 10  |-  ( (
<. A ,  B >.pprod (  _I  , Singleton ) z  /\  z (Singleton  o. Img ) y
)  <->  ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y ) )
3938exbii 1639 . . . . . . . . 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 4706 . . . . . . . . . . 11  |-  <. A ,  { B } >.  e.  _V
41 breq1 4445 . . . . . . . . . . 11  |-  ( z  =  <. A ,  { B } >.  ->  ( z (Singleton  o. Img ) y  <->  <. A ,  { B } >. (Singleton  o. Img ) y ) )
4240, 41ceqsexv 3145 . . . . . . . . . 10  |-  ( E. z ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y )  <->  <. A ,  { B } >. (Singleton  o. Img ) y )
4340, 14brco 5166 . . . . . . . . . 10  |-  ( <. A ,  { B } >. (Singleton  o. Img ) y  <->  E. x ( <. A ,  { B } >.Img x  /\  xSingleton y ) )
4416, 31, 12brimg 29152 . . . . . . . . . . . . 13  |-  ( <. A ,  { B } >.Img x  <->  x  =  ( A " { B } ) )
4512, 14brsingle 29132 . . . . . . . . . . . . 13  |-  ( xSingleton
y  <->  y  =  {
x } )
4644, 45anbi12i 697 . . . . . . . . . . . 12  |-  ( (
<. A ,  { B } >.Img x  /\  xSingleton y )  <->  ( x  =  ( A " { B } )  /\  y  =  { x } ) )
4746exbii 1639 . . . . . . . . . . 11  |-  ( E. x ( <. A ,  { B } >.Img x  /\  xSingleton y )  <->  E. x
( x  =  ( A " { B } )  /\  y  =  { x } ) )
48 imaexg 6713 . . . . . . . . . . . . 13  |-  ( A  e.  _V  ->  ( A " { B }
)  e.  _V )
4916, 48ax-mp 5 . . . . . . . . . . . 12  |-  ( A
" { B }
)  e.  _V
50 sneq 4032 . . . . . . . . . . . . 13  |-  ( x  =  ( A " { B } )  ->  { x }  =  { ( A " { B } ) } )
5150eqeq2d 2476 . . . . . . . . . . . 12  |-  ( x  =  ( A " { B } )  -> 
( y  =  {
x }  <->  y  =  { ( A " { B } ) } ) )
5249, 51ceqsexv 3145 . . . . . . . . . . 11  |-  ( E. x ( x  =  ( A " { B } )  /\  y  =  { x } )  <-> 
y  =  { ( A " { B } ) } )
5347, 52bitri 249 . . . . . . . . . 10  |-  ( E. x ( <. A ,  { B } >.Img x  /\  xSingleton y )  <->  y  =  { ( A " { B } ) } )
5442, 43, 533bitri 271 . . . . . . . . 9  |-  ( E. z ( z  = 
<. A ,  { B } >.  /\  z (Singleton  o. Img ) y )  <->  y  =  { ( A " { B } ) } )
5515, 39, 543bitri 271 . . . . . . . 8  |-  ( <. A ,  B >. ( (Singleton  o. Img )  o. pprod (  _I  , Singleton ) ) y  <-> 
y  =  { ( A " { B } ) } )
56 eqid 2462 . . . . . . . . 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 5024 . . . . . . . . . 10  |-  ( y ( _V  X.  _V ) x  <->  ( y  e. 
_V  /\  x  e.  _V ) )
5814, 12, 57mpbir2an 913 . . . . . . . . 9  |-  y ( _V  X.  _V )
x
59 epel 4789 . . . . . . . . . . 11  |-  ( z  _E  y  <->  z  e.  y )
6059anbi1i 695 . . . . . . . . . 10  |-  ( ( z  _E  y  /\  z  e.  Singletons )  <->  ( z  e.  y  /\  z  e. 
Singletons ) )
6114brres 5273 . . . . . . . . . 10  |-  ( z (  _E  |`  Singletons ) y  <->  ( z  _E  y  /\  z  e. 
Singletons ) )
62 elin 3682 . . . . . . . . . 10  |-  ( z  e.  ( y  i^i  Singletons
)  <->  ( z  e.  y  /\  z  e.  Singletons
) )
6360, 61, 623bitr4ri 278 . . . . . . . . 9  |-  ( z  e.  ( y  i^i  Singletons
)  <->  z (  _E  |` 
Singletons ) y )
6414, 12, 56, 58, 63brtxpsd3 29111 . . . . . . . 8  |-  ( y ( ( _V  X.  _V )  \  ran  (
( _V  (x)  _E  )(++) ( (  _E  |`  Singletons )  (x)  _V ) ) ) x  <-> 
x  =  ( y  i^i  Singletons ) )
6555, 64anbi12i 697 . . . . . . 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 1639 . . . . . 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 3688 . . . . . . . 8  |-  ( y  =  { ( A
" { B }
) }  ->  (
y  i^i  Singletons )  =  ( { ( A " { B } ) }  i^i  Singletons ) )
6867eqeq2d 2476 . . . . . . 7  |-  ( y  =  { ( A
" { B }
) }  ->  (
x  =  ( y  i^i  Singletons )  <->  x  =  ( { ( A " { B } ) }  i^i  Singletons ) ) )
691, 68ceqsexv 3145 . . . . . 6  |-  ( E. y ( y  =  { ( A " { B } ) }  /\  x  =  ( y  i^i  Singletons ) )  <->  x  =  ( { ( A " { B } ) }  i^i  Singletons ) )
7013, 66, 693bitri 271 . . . . 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 5166 . . . . . 6  |-  ( x ( Bigcup  o.  Bigcup ) C  <->  E. y ( x Bigcup y  /\  y Bigcup C ) )
7214brbigcup 29113 . . . . . . . . 9  |-  ( x
Bigcup y  <->  U. x  =  y )
73 eqcom 2471 . . . . . . . . 9  |-  ( U. x  =  y  <->  y  =  U. x )
7472, 73bitri 249 . . . . . . . 8  |-  ( x
Bigcup y  <->  y  =  U. x )
7510brbigcup 29113 . . . . . . . . 9  |-  ( y
Bigcup C  <->  U. y  =  C )
76 eqcom 2471 . . . . . . . . 9  |-  ( U. y  =  C  <->  C  =  U. y )
7775, 76bitri 249 . . . . . . . 8  |-  ( y
Bigcup C  <->  C  =  U. y )
7874, 77anbi12i 697 . . . . . . 7  |-  ( ( x Bigcup y  /\  y Bigcup C )  <->  ( y  =  U. x  /\  C  =  U. y ) )
7978exbii 1639 . . . . . 6  |-  ( E. y ( x Bigcup y  /\  y Bigcup C )  <->  E. y ( y  = 
U. x  /\  C  =  U. y ) )
8012uniex 6573 . . . . . . 7  |-  U. x  e.  _V
81 unieq 4248 . . . . . . . 8  |-  ( y  =  U. x  ->  U. y  =  U. U. x )
8281eqeq2d 2476 . . . . . . 7  |-  ( y  =  U. x  -> 
( C  =  U. y 
<->  C  =  U. U. x ) )
8380, 82ceqsexv 3145 . . . . . 6  |-  ( E. y ( y  = 
U. x  /\  C  =  U. y )  <->  C  =  U. U. x )
8471, 79, 833bitri 271 . . . . 5  |-  ( x ( Bigcup  o.  Bigcup ) C  <-> 
C  =  U. U. x )
8570, 84anbi12i 697 . . . 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 1639 . . 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 271 . 2  |-  ( <. A ,  B >.Apply C  <->  E. x ( x  =  ( { ( A
" { B }
) }  i^i  Singletons )  /\  C  =  U. U. x ) )
88 dffv5 29139 . . 3  |-  ( A `
 B )  = 
U. U. ( { ( A " { B } ) }  i^i  Singletons )
8988eqeq2i 2480 . 2  |-  ( C  =  ( A `  B )  <->  C  =  U. U. ( { ( A " { B } ) }  i^i  Singletons ) )
906, 87, 893bitr4i 277 1  |-  ( <. A ,  B >.Apply C  <-> 
C  =  ( A `
 B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3108    \ cdif 3468    i^i cin 3470   {csn 4022   <.cop 4028   U.cuni 4240   class class class wbr 4442    _E cep 4784    _I cid 4785    X. cxp 4992   ran crn 4995    |` cres 4996   "cima 4997    o. ccom 4998   ` cfv 5581  (++)csymdif 29032    (x) ctxp 29044  pprodcpprod 29045   Bigcupcbigcup 29048  Singletoncsingle 29052   Singletonscsingles 29053  Imgcimg 29056  Applycapply 29059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-eprel 4786  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-1st 6776  df-2nd 6777  df-symdif 29033  df-txp 29068  df-pprod 29069  df-bigcup 29072  df-singleton 29076  df-singles 29077  df-image 29078  df-cart 29079  df-img 29080  df-apply 29087
This theorem is referenced by:  dfrdg4  29165  tfrqfree  29166
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