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Theorem brimg 30753
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 30681 . . 3  |- Img  =  (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
)
21breqi 4422 . 2  |-  ( <. A ,  B >.Img C  <->  <. A ,  B >. (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
) C )
3 opex 4678 . . . 4  |-  <. A ,  B >.  e.  _V
4 brimg.3 . . . 4  |-  C  e. 
_V
53, 4brco 5024 . . 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 3060 . . . . . 6  |-  a  e. 
_V
96, 7, 8brcart 30748 . . . . 5  |-  ( <. A ,  B >.Cart a  <-> 
a  =  ( A  X.  B ) )
109anbi1i 706 . . . 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 1729 . . 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 6622 . . . 4  |-  ( A  X.  B )  e. 
_V
13 breq1 4419 . . . 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 3096 . . 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 279 . 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 30742 . . 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 1845 . . . . . . . 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 659 . . . . . . . . . . 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 659 . . . . . . . . . . . . 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 811 . . . . . . . . . . . . 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 257 . . . . . . . . . . . 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 705 . . . . . . . . . . 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 257 . . . . . . . . . 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 1730 . . . . . . . . 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 1938 . . . . . . . . 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 4678 . . . . . . . . . . 11  |-  <. a ,  b >.  e.  _V
27 breq1 4419 . . . . . . . . . . . . 13  |-  ( p  =  <. a ,  b
>.  ->  ( p ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x  <->  <. a ,  b >. ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
2827anbi2d 715 . . . . . . . . . . . 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 715 . . . . . . . . . . 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 3096 . . . . . . . . . 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 1729 . . . . . . . . 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 279 . . . . . . . 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 4417 . . . . . . . . . 10  |-  ( b A x  <->  <. b ,  x >.  e.  A
)
34 risset 2927 . . . . . . . . . . 11  |-  ( <.
b ,  x >.  e.  A  <->  E. a  e.  A  a  =  <. b ,  x >. )
35 vex 3060 . . . . . . . . . . . . . . . 16  |-  b  e. 
_V
3635brres 5130 . . . . . . . . . . . . . . 15  |-  ( a ( 1st  |`  ( _V  X.  _V ) ) b  <->  ( a 1st b  /\  a  e.  ( _V  X.  _V ) ) )
37 df-br 4417 . . . . . . . . . . . . . . 15  |-  ( a ( 1st  |`  ( _V  X.  _V ) ) b  <->  <. a ,  b
>.  e.  ( 1st  |`  ( _V  X.  _V ) ) )
38 ancom 456 . . . . . . . . . . . . . . 15  |-  ( ( a 1st b  /\  a  e.  ( _V  X.  _V ) )  <->  ( a  e.  ( _V  X.  _V )  /\  a 1st b
) )
3936, 37, 383bitr3ri 284 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ( _V 
X.  _V )  /\  a 1st b )  <->  <. a ,  b >.  e.  ( 1st  |`  ( _V  X.  _V ) ) )
4039anbi2i 705 . . . . . . . . . . . . 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 4912 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( _V  X.  _V )  <->  E. p E. q 
a  =  <. p ,  q >. )
4241anbi1i 706 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  =  <. p ,  q >.  /\  (
p  =  b  /\  q  =  x )
)  <->  ( ( p  =  b  /\  q  =  x )  /\  a  =  <. p ,  q
>. ) )
45 breq1 4419 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  <. p ,  q
>.  ->  ( a 1st b  <->  <. p ,  q
>. 1st b ) )
46 opeq1 4180 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  =  <. p ,  q
>.  ->  <. a ,  b
>.  =  <. <. p ,  q >. ,  b
>. )
4746breq1d 4426 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  <. p ,  q
>.  ->  ( <. a ,  b >. ( 2nd  o.  1st ) x  <->  <. <. p ,  q
>. ,  b >. ( 2nd  o.  1st )
x ) )
4845, 47anbi12d 722 . . . . . . . . . . . . . . . . . . . 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 3060 . . . . . . . . . . . . . . . . . . . . . . 23  |-  p  e. 
_V
50 vex 3060 . . . . . . . . . . . . . . . . . . . . . . 23  |-  q  e. 
_V
5149, 50, 35br1steq 30463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
p ,  q >. 1st b  <->  b  =  p )
52 equcom 1873 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  =  p  <->  p  =  b )
5351, 52bitri 257 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
p ,  q >. 1st b  <->  p  =  b
)
54 opex 4678 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  <. <. p ,  q >. ,  b
>.  e.  _V
55 vex 3060 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  x  e. 
_V
5654, 55brco 5024 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <->  E. a ( <. <. p ,  q >. ,  b
>. 1st a  /\  a 2nd x ) )
57 opex 4678 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  <. p ,  q >.  e.  _V
5857, 35, 8br1steq 30463 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( <. <. p ,  q >. ,  b >. 1st a  <->  a  =  <. p ,  q
>. )
5958anbi1i 706 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
<. <. p ,  q
>. ,  b >. 1st a  /\  a 2nd x )  <->  ( a  =  <. p ,  q
>.  /\  a 2nd x
) )
6059exbii 1729 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E. a ( <. <. p ,  q >. ,  b
>. 1st a  /\  a 2nd x )  <->  E. a
( a  =  <. p ,  q >.  /\  a 2nd x ) )
6156, 60bitri 257 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <->  E. a ( a  = 
<. p ,  q >.  /\  a 2nd x ) )
62 breq1 4419 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( a  =  <. p ,  q
>.  ->  ( a 2nd x  <->  <. p ,  q
>. 2nd x ) )
6357, 62ceqsexv 3096 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E. a ( a  = 
<. p ,  q >.  /\  a 2nd x )  <->  <. p ,  q >. 2nd x )
6449, 50, 55br2ndeq 30464 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
p ,  q >. 2nd x  <->  x  =  q
)
6563, 64bitri 257 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. a ( a  = 
<. p ,  q >.  /\  a 2nd x )  <-> 
x  =  q )
66 equcom 1873 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  q  <->  q  =  x )
6761, 65, 663bitri 279 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <-> 
q  =  x )
6853, 67anbi12i 708 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. p ,  q >. 1st b  /\  <. <. p ,  q >. ,  b
>. ( 2nd  o.  1st ) x )  <->  ( p  =  b  /\  q  =  x ) )
6948, 68syl6bb 269 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  <. p ,  q
>.  ->  ( ( a 1st b  /\  <. a ,  b >. ( 2nd  o.  1st ) x )  <->  ( p  =  b  /\  q  =  x ) ) )
7069pm5.32i 647 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( a  =  <. p ,  q >.  /\  (
p  =  b  /\  q  =  x )
) )
71 df-3an 993 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )  <->  ( (
p  =  b  /\  q  =  x )  /\  a  =  <. p ,  q >. )
)
7244, 70, 713bitr4i 285 . . . . . . . . . . . . . . . . 17  |-  ( ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )
)
73722exbii 1730 . . . . . . . . . . . . . . . 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 1842 . . . . . . . . . . . . . . . 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 4180 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  b  ->  <. p ,  q >.  =  <. b ,  q >. )
7675eqeq2d 2472 . . . . . . . . . . . . . . . . 17  |-  ( p  =  b  ->  (
a  =  <. p ,  q >.  <->  a  =  <. b ,  q >.
) )
77 opeq2 4181 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  x  ->  <. b ,  q >.  =  <. b ,  x >. )
7877eqeq2d 2472 . . . . . . . . . . . . . . . . 17  |-  ( q  =  x  ->  (
a  =  <. b ,  q >.  <->  a  =  <. b ,  x >. ) )
7935, 55, 76, 78ceqsex2v 3099 . . . . . . . . . . . . . . . 16  |-  ( E. p E. q ( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )  <->  a  =  <. b ,  x >. )
8073, 74, 793bitr3ri 284 . . . . . . . . . . . . . . 15  |-  ( a  =  <. b ,  x >.  <-> 
( E. p E. q  a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) ) )
8142, 43, 803bitr4ri 286 . . . . . . . . . . . . . 14  |-  ( a  =  <. b ,  x >.  <-> 
( ( a  e.  ( _V  X.  _V )  /\  a 1st b
)  /\  <. a ,  b >. ( 2nd  o.  1st ) x ) )
82 ancom 456 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 13  |-  ( a  =  <. b ,  x >.  <-> 
( <. a ,  b
>. ( 2nd  o.  1st ) x  /\  (
a  e.  ( _V 
X.  _V )  /\  a 1st b ) ) )
8455brres 5130 . . . . . . . . . . . . 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 285 . . . . . . . . . . . 12  |-  ( a  =  <. b ,  x >.  <->  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
8685rexbii 2901 . . . . . . . . . . 11  |-  ( E. a  e.  A  a  =  <. b ,  x >.  <->  E. a  e.  A  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
8734, 86bitri 257 . . . . . . . . . 10  |-  ( <.
b ,  x >.  e.  A  <->  E. a  e.  A  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
88 df-rex 2755 . . . . . . . . . 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 279 . . . . . . . . 9  |-  ( b A x  <->  E. a
( a  e.  A  /\  <. a ,  b
>. ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
9089anbi2i 705 . . . . . . . 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 286 . . . . . . 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 1729 . . . . . 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 5193 . . . . . 6  |-  ( x  e.  ( A " B )  <->  E. b
( b  e.  B  /\  b A x ) )
9455elima2 5193 . . . . . . 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 4870 . . . . . . . . . 10  |-  ( p  e.  ( A  X.  B )  <->  E. a E. b ( p  = 
<. a ,  b >.  /\  ( a  e.  A  /\  b  e.  B
) ) )
9695anbi1i 706 . . . . . . . . 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 1842 . . . . . . . . 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 260 . . . . . . . 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 1729 . . . . . . 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 1942 . . . . . . . 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 1942 . . . . . . . 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 257 . . . . . . 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 279 . . . . . 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 286 . . . . 5  |-  ( x  e.  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <-> 
x  e.  ( A
" B ) )
105104eqriv 2459 . . . 4  |-  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  =  ( A " B )
106105eqeq2i 2474 . . 3  |-  ( C  =  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <-> 
C  =  ( A
" B ) )
10716, 106bitri 257 . 2  |-  ( ( A  X.  B )Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C  <-> 
C  =  ( A
" B ) )
1082, 15, 1073bitri 279 1  |-  ( <. A ,  B >.Img C  <-> 
C  =  ( A
" B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455   E.wex 1674    e. wcel 1898   E.wrex 2750   _Vcvv 3057   <.cop 3986   class class class wbr 4416    X. cxp 4851    |` cres 4855   "cima 4856    o. ccom 4857   1stc1st 6818   2ndc2nd 6819  Imagecimage 30655  Cartccart 30656  Imgcimg 30657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-symdif 3675  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-eprel 4764  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6820  df-2nd 6821  df-txp 30669  df-pprod 30670  df-image 30679  df-cart 30680  df-img 30681
This theorem is referenced by:  brapply  30754  dfrdg4  30767
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