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Theorem brimg 27968
Description: The binary relationship form of the Img function. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brimg.1  |-  A  e. 
_V
brimg.2  |-  B  e. 
_V
brimg.3  |-  C  e. 
_V
Assertion
Ref Expression
brimg  |-  ( <. A ,  B >.Img C  <-> 
C  =  ( A
" B ) )

Proof of Theorem brimg
Dummy variables  a 
b  p  q  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-img 27896 . . 3  |- Img  =  (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
)
21breqi 4298 . 2  |-  ( <. A ,  B >.Img C  <->  <. A ,  B >. (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
) C )
3 opex 4556 . . . 4  |-  <. A ,  B >.  e.  _V
4 brimg.3 . . . 4  |-  C  e. 
_V
53, 4brco 5010 . . 3  |-  ( <. A ,  B >. (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
) C  <->  E. a
( <. A ,  B >.Cart a  /\  aImage (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C ) )
6 brimg.1 . . . . . 6  |-  A  e. 
_V
7 brimg.2 . . . . . 6  |-  B  e. 
_V
8 vex 2975 . . . . . 6  |-  a  e. 
_V
96, 7, 8brcart 27963 . . . . 5  |-  ( <. A ,  B >.Cart a  <-> 
a  =  ( A  X.  B ) )
109anbi1i 695 . . . 4  |-  ( (
<. A ,  B >.Cart a  /\  aImage ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C )  <->  ( a  =  ( A  X.  B )  /\  aImage ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C ) )
1110exbii 1634 . . 3  |-  ( E. a ( <. A ,  B >.Cart a  /\  aImage ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C )  <->  E. a ( a  =  ( A  X.  B )  /\  aImage ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C ) )
126, 7xpex 6508 . . . 4  |-  ( A  X.  B )  e. 
_V
13 breq1 4295 . . . 4  |-  ( a  =  ( A  X.  B )  ->  (
aImage ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C  <-> 
( A  X.  B
)Image ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C ) )
1412, 13ceqsexv 3009 . . 3  |-  ( E. a ( a  =  ( A  X.  B
)  /\  aImage (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C )  <->  ( A  X.  B )Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C )
155, 11, 143bitri 271 . 2  |-  ( <. A ,  B >. (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
) C  <->  ( A  X.  B )Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C )
1612, 4brimage 27957 . . 3  |-  ( ( A  X.  B )Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C  <-> 
C  =  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) ) )
17 19.42v 1924 . . . . . . . 8  |-  ( E. a ( b  e.  B  /\  ( a  e.  A  /\  <. a ,  b >. (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )  <-> 
( b  e.  B  /\  E. a ( a  e.  A  /\  <. a ,  b >. (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
18 anass 649 . . . . . . . . . . 11  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  ( p  =  <. a ,  b
>.  /\  ( ( a  e.  A  /\  b  e.  B )  /\  p
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
19 anass 649 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  ( a  e.  A  /\  (
b  e.  B  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
20 an12 795 . . . . . . . . . . . . 13  |-  ( ( a  e.  A  /\  ( b  e.  B  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )  <->  ( b  e.  B  /\  (
a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
2119, 20bitri 249 . . . . . . . . . . . 12  |-  ( ( ( a  e.  A  /\  b  e.  B
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  ( b  e.  B  /\  (
a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
2221anbi2i 694 . . . . . . . . . . 11  |-  ( ( p  =  <. a ,  b >.  /\  (
( a  e.  A  /\  b  e.  B
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )  <-> 
( p  =  <. a ,  b >.  /\  (
b  e.  B  /\  ( a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) ) )
2318, 22bitri 249 . . . . . . . . . 10  |-  ( ( ( p  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  ( p  =  <. a ,  b
>.  /\  ( b  e.  B  /\  ( a  e.  A  /\  p
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) ) )
24232exbii 1635 . . . . . . . . 9  |-  ( E. p E. a ( ( p  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  E. p E. a ( p  = 
<. a ,  b >.  /\  ( b  e.  B  /\  ( a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) ) )
25 excom 1787 . . . . . . . . 9  |-  ( E. p E. a ( p  =  <. a ,  b >.  /\  (
b  e.  B  /\  ( a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )  <->  E. a E. p ( p  = 
<. a ,  b >.  /\  ( b  e.  B  /\  ( a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) ) )
26 opex 4556 . . . . . . . . . . 11  |-  <. a ,  b >.  e.  _V
27 breq1 4295 . . . . . . . . . . . . 13  |-  ( p  =  <. a ,  b
>.  ->  ( p ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x  <->  <. a ,  b >. ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
2827anbi2d 703 . . . . . . . . . . . 12  |-  ( p  =  <. a ,  b
>.  ->  ( ( a  e.  A  /\  p
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  ( a  e.  A  /\  <. a ,  b >. (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
2928anbi2d 703 . . . . . . . . . . 11  |-  ( p  =  <. a ,  b
>.  ->  ( ( b  e.  B  /\  (
a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )  <->  ( b  e.  B  /\  (
a  e.  A  /\  <.
a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) ) )
3026, 29ceqsexv 3009 . . . . . . . . . 10  |-  ( E. p ( p  = 
<. a ,  b >.  /\  ( b  e.  B  /\  ( a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )  <->  ( b  e.  B  /\  (
a  e.  A  /\  <.
a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
3130exbii 1634 . . . . . . . . 9  |-  ( E. a E. p ( p  =  <. a ,  b >.  /\  (
b  e.  B  /\  ( a  e.  A  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )  <->  E. a
( b  e.  B  /\  ( a  e.  A  /\  <. a ,  b
>. ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
3224, 25, 313bitri 271 . . . . . . . 8  |-  ( E. p E. a ( ( p  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  E. a
( b  e.  B  /\  ( a  e.  A  /\  <. a ,  b
>. ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
33 df-br 4293 . . . . . . . . . 10  |-  ( b A x  <->  <. b ,  x >.  e.  A
)
34 risset 2763 . . . . . . . . . . 11  |-  ( <.
b ,  x >.  e.  A  <->  E. a  e.  A  a  =  <. b ,  x >. )
35 vex 2975 . . . . . . . . . . . . . . . 16  |-  b  e. 
_V
3635brres 5117 . . . . . . . . . . . . . . 15  |-  ( a ( 1st  |`  ( _V  X.  _V ) ) b  <->  ( a 1st b  /\  a  e.  ( _V  X.  _V ) ) )
37 df-br 4293 . . . . . . . . . . . . . . 15  |-  ( a ( 1st  |`  ( _V  X.  _V ) ) b  <->  <. a ,  b
>.  e.  ( 1st  |`  ( _V  X.  _V ) ) )
38 ancom 450 . . . . . . . . . . . . . . 15  |-  ( ( a 1st b  /\  a  e.  ( _V  X.  _V ) )  <->  ( a  e.  ( _V  X.  _V )  /\  a 1st b
) )
3936, 37, 383bitr3ri 276 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ( _V 
X.  _V )  /\  a 1st b )  <->  <. a ,  b >.  e.  ( 1st  |`  ( _V  X.  _V ) ) )
4039anbi2i 694 . . . . . . . . . . . . 13  |-  ( (
<. a ,  b >.
( 2nd  o.  1st ) x  /\  (
a  e.  ( _V 
X.  _V )  /\  a 1st b ) )  <->  ( <. a ,  b >. ( 2nd  o.  1st ) x  /\  <. a ,  b
>.  e.  ( 1st  |`  ( _V  X.  _V ) ) ) )
41 elvv 4897 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( _V  X.  _V )  <->  E. p E. q 
a  =  <. p ,  q >. )
4241anbi1i 695 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( E. p E. q  a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) ) )
43 anass 649 . . . . . . . . . . . . . . 15  |-  ( ( ( a  e.  ( _V  X.  _V )  /\  a 1st b )  /\  <. a ,  b
>. ( 2nd  o.  1st ) x )  <->  ( a  e.  ( _V  X.  _V )  /\  ( a 1st b  /\  <. a ,  b >. ( 2nd  o.  1st ) x ) ) )
44 ancom 450 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  =  <. p ,  q >.  /\  (
p  =  b  /\  q  =  x )
)  <->  ( ( p  =  b  /\  q  =  x )  /\  a  =  <. p ,  q
>. ) )
45 breq1 4295 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  <. p ,  q
>.  ->  ( a 1st b  <->  <. p ,  q
>. 1st b ) )
46 opeq1 4059 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  =  <. p ,  q
>.  ->  <. a ,  b
>.  =  <. <. p ,  q >. ,  b
>. )
4746breq1d 4302 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  <. p ,  q
>.  ->  ( <. a ,  b >. ( 2nd  o.  1st ) x  <->  <. <. p ,  q
>. ,  b >. ( 2nd  o.  1st )
x ) )
4845, 47anbi12d 710 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  <. p ,  q
>.  ->  ( ( a 1st b  /\  <. a ,  b >. ( 2nd  o.  1st ) x )  <->  ( <. p ,  q >. 1st b  /\  <. <. p ,  q
>. ,  b >. ( 2nd  o.  1st )
x ) ) )
49 vex 2975 . . . . . . . . . . . . . . . . . . . . . . 23  |-  p  e. 
_V
50 vex 2975 . . . . . . . . . . . . . . . . . . . . . . 23  |-  q  e. 
_V
5149, 50, 35br1steq 27585 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
p ,  q >. 1st b  <->  b  =  p )
52 equcom 1732 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  =  p  <->  p  =  b )
5351, 52bitri 249 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
p ,  q >. 1st b  <->  p  =  b
)
54 opex 4556 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  <. <. p ,  q >. ,  b
>.  e.  _V
55 vex 2975 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  x  e. 
_V
5654, 55brco 5010 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <->  E. a ( <. <. p ,  q >. ,  b
>. 1st a  /\  a 2nd x ) )
57 opex 4556 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  <. p ,  q >.  e.  _V
5857, 35, 8br1steq 27585 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( <. <. p ,  q >. ,  b >. 1st a  <->  a  =  <. p ,  q
>. )
5958anbi1i 695 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
<. <. p ,  q
>. ,  b >. 1st a  /\  a 2nd x )  <->  ( a  =  <. p ,  q
>.  /\  a 2nd x
) )
6059exbii 1634 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E. a ( <. <. p ,  q >. ,  b
>. 1st a  /\  a 2nd x )  <->  E. a
( a  =  <. p ,  q >.  /\  a 2nd x ) )
6156, 60bitri 249 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <->  E. a ( a  = 
<. p ,  q >.  /\  a 2nd x ) )
62 breq1 4295 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( a  =  <. p ,  q
>.  ->  ( a 2nd x  <->  <. p ,  q
>. 2nd x ) )
6357, 62ceqsexv 3009 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E. a ( a  = 
<. p ,  q >.  /\  a 2nd x )  <->  <. p ,  q >. 2nd x )
6449, 50, 55br2ndeq 27586 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
p ,  q >. 2nd x  <->  x  =  q
)
6563, 64bitri 249 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. a ( a  = 
<. p ,  q >.  /\  a 2nd x )  <-> 
x  =  q )
66 equcom 1732 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  q  <->  q  =  x )
6761, 65, 663bitri 271 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <-> 
q  =  x )
6853, 67anbi12i 697 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. p ,  q >. 1st b  /\  <. <. p ,  q >. ,  b
>. ( 2nd  o.  1st ) x )  <->  ( p  =  b  /\  q  =  x ) )
6948, 68syl6bb 261 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  <. p ,  q
>.  ->  ( ( a 1st b  /\  <. a ,  b >. ( 2nd  o.  1st ) x )  <->  ( p  =  b  /\  q  =  x ) ) )
7069pm5.32i 637 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( a  =  <. p ,  q >.  /\  (
p  =  b  /\  q  =  x )
) )
71 df-3an 967 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )  <->  ( (
p  =  b  /\  q  =  x )  /\  a  =  <. p ,  q >. )
)
7244, 70, 713bitr4i 277 . . . . . . . . . . . . . . . . 17  |-  ( ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )
)
73722exbii 1635 . . . . . . . . . . . . . . . 16  |-  ( E. p E. q ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <->  E. p E. q ( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. ) )
74 19.41vv 1921 . . . . . . . . . . . . . . . 16  |-  ( E. p E. q ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( E. p E. q  a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) ) )
75 opeq1 4059 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  b  ->  <. p ,  q >.  =  <. b ,  q >. )
7675eqeq2d 2454 . . . . . . . . . . . . . . . . 17  |-  ( p  =  b  ->  (
a  =  <. p ,  q >.  <->  a  =  <. b ,  q >.
) )
77 opeq2 4060 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  x  ->  <. b ,  q >.  =  <. b ,  x >. )
7877eqeq2d 2454 . . . . . . . . . . . . . . . . 17  |-  ( q  =  x  ->  (
a  =  <. b ,  q >.  <->  a  =  <. b ,  x >. ) )
7935, 55, 76, 78ceqsex2v 3011 . . . . . . . . . . . . . . . 16  |-  ( E. p E. q ( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )  <->  a  =  <. b ,  x >. )
8073, 74, 793bitr3ri 276 . . . . . . . . . . . . . . 15  |-  ( a  =  <. b ,  x >.  <-> 
( E. p E. q  a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) ) )
8142, 43, 803bitr4ri 278 . . . . . . . . . . . . . 14  |-  ( a  =  <. b ,  x >.  <-> 
( ( a  e.  ( _V  X.  _V )  /\  a 1st b
)  /\  <. a ,  b >. ( 2nd  o.  1st ) x ) )
82 ancom 450 . . . . . . . . . . . . . 14  |-  ( ( ( a  e.  ( _V  X.  _V )  /\  a 1st b )  /\  <. a ,  b
>. ( 2nd  o.  1st ) x )  <->  ( <. a ,  b >. ( 2nd  o.  1st ) x  /\  ( a  e.  ( _V  X.  _V )  /\  a 1st b
) ) )
8381, 82bitri 249 . . . . . . . . . . . . 13  |-  ( a  =  <. b ,  x >.  <-> 
( <. a ,  b
>. ( 2nd  o.  1st ) x  /\  (
a  e.  ( _V 
X.  _V )  /\  a 1st b ) ) )
8455brres 5117 . . . . . . . . . . . . 13  |-  ( <.
a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x  <-> 
( <. a ,  b
>. ( 2nd  o.  1st ) x  /\  <. a ,  b >.  e.  ( 1st  |`  ( _V  X.  _V ) ) ) )
8540, 83, 843bitr4i 277 . . . . . . . . . . . 12  |-  ( a  =  <. b ,  x >.  <->  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
8685rexbii 2740 . . . . . . . . . . 11  |-  ( E. a  e.  A  a  =  <. b ,  x >.  <->  E. a  e.  A  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
8734, 86bitri 249 . . . . . . . . . 10  |-  ( <.
b ,  x >.  e.  A  <->  E. a  e.  A  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
88 df-rex 2721 . . . . . . . . . 10  |-  ( E. a  e.  A  <. a ,  b >. (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x  <->  E. a
( a  e.  A  /\  <. a ,  b
>. ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
8933, 87, 883bitri 271 . . . . . . . . 9  |-  ( b A x  <->  E. a
( a  e.  A  /\  <. a ,  b
>. ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
9089anbi2i 694 . . . . . . . 8  |-  ( ( b  e.  B  /\  b A x )  <->  ( b  e.  B  /\  E. a
( a  e.  A  /\  <. a ,  b
>. ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) ) )
9117, 32, 903bitr4ri 278 . . . . . . 7  |-  ( ( b  e.  B  /\  b A x )  <->  E. p E. a ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
9291exbii 1634 . . . . . 6  |-  ( E. b ( b  e.  B  /\  b A x )  <->  E. b E. p E. a ( ( p  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
9355elima2 5175 . . . . . 6  |-  ( x  e.  ( A " B )  <->  E. b
( b  e.  B  /\  b A x ) )
9455elima2 5175 . . . . . . 7  |-  ( x  e.  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <->  E. p ( p  e.  ( A  X.  B
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
95 elxp 4857 . . . . . . . . . 10  |-  ( p  e.  ( A  X.  B )  <->  E. a E. b ( p  = 
<. a ,  b >.  /\  ( a  e.  A  /\  b  e.  B
) ) )
9695anbi1i 695 . . . . . . . . 9  |-  ( ( p  e.  ( A  X.  B )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  ( E. a E. b ( p  = 
<. a ,  b >.  /\  ( a  e.  A  /\  b  e.  B
) )  /\  p
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
97 19.41vv 1921 . . . . . . . . 9  |-  ( E. a E. b ( ( p  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  ( E. a E. b ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
9896, 97bitr4i 252 . . . . . . . 8  |-  ( ( p  e.  ( A  X.  B )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  E. a E. b
( ( p  = 
<. a ,  b >.  /\  ( a  e.  A  /\  b  e.  B
) )  /\  p
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
9998exbii 1634 . . . . . . 7  |-  ( E. p ( p  e.  ( A  X.  B
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  E. p E. a E. b ( ( p  =  <. a ,  b >.  /\  (
a  e.  A  /\  b  e.  B )
)  /\  p (
( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
100 exrot3 1790 . . . . . . . 8  |-  ( E. p E. a E. b ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  E. a E. b E. p ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
101 exrot3 1790 . . . . . . . 8  |-  ( E. a E. b E. p ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  E. b E. p E. a ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
102100, 101bitri 249 . . . . . . 7  |-  ( E. p E. a E. b ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )  <->  E. b E. p E. a ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
10394, 99, 1023bitri 271 . . . . . 6  |-  ( x  e.  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <->  E. b E. p E. a ( ( p  =  <. a ,  b
>.  /\  ( a  e.  A  /\  b  e.  B ) )  /\  p ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
10492, 93, 1033bitr4ri 278 . . . . 5  |-  ( x  e.  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <-> 
x  e.  ( A
" B ) )
105104eqriv 2440 . . . 4  |-  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  =  ( A " B )
106105eqeq2i 2453 . . 3  |-  ( C  =  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <-> 
C  =  ( A
" B ) )
10716, 106bitri 249 . 2  |-  ( ( A  X.  B )Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C  <-> 
C  =  ( A
" B ) )
1082, 15, 1073bitri 271 1  |-  ( <. A ,  B >.Img C  <-> 
C  =  ( A
" B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2716   _Vcvv 2972   <.cop 3883   class class class wbr 4292    X. cxp 4838    |` cres 4842   "cima 4843    o. ccom 4844   1stc1st 6575   2ndc2nd 6576  Imagecimage 27870  Cartccart 27871  Imgcimg 27872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-eprel 4632  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426  df-1st 6577  df-2nd 6578  df-symdif 27849  df-txp 27884  df-pprod 27885  df-image 27894  df-cart 27895  df-img 27896
This theorem is referenced by:  brapply  27969  dfrdg4  27981
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