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Theorem brimg 25727
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 25657 . . 3  |- Img  =  (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
)
21breqi 4205 . 2  |-  ( <. A ,  B >.Img C  <->  <. A ,  B >. (Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) )  o. Cart
) C )
3 opex 4414 . . . 4  |-  <. A ,  B >.  e.  _V
4 brimg.3 . . . 4  |-  C  e. 
_V
53, 4brco 5029 . . 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 2946 . . . . . 6  |-  a  e. 
_V
96, 7, 8brcart 25722 . . . . 5  |-  ( <. A ,  B >.Cart a  <-> 
a  =  ( A  X.  B ) )
109anbi1i 677 . . . 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 1592 . . 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 4976 . . . 4  |-  ( A  X.  B )  e. 
_V
13 breq1 4202 . . . 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 2978 . . 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 263 . 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 25716 . . 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 1928 . . . . . . . 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 631 . . . . . . . . . . 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 631 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 241 . . . . . . . . . . . 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 676 . . . . . . . . . . 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 241 . . . . . . . . . 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 1593 . . . . . . . . 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 1756 . . . . . . . . 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 4414 . . . . . . . . . . 11  |-  <. a ,  b >.  e.  _V
27 breq1 4202 . . . . . . . . . . . . 13  |-  ( p  =  <. a ,  b
>.  ->  ( p ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x  <->  <. a ,  b >. ( ( 2nd 
o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
2827anbi2d 685 . . . . . . . . . . . 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 685 . . . . . . . . . . 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 2978 . . . . . . . . . 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 1592 . . . . . . . . 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 263 . . . . . . . 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 4200 . . . . . . . . . 10  |-  ( b A x  <->  <. b ,  x >.  e.  A
)
34 risset 2740 . . . . . . . . . . 11  |-  ( <.
b ,  x >.  e.  A  <->  E. a  e.  A  a  =  <. b ,  x >. )
35 vex 2946 . . . . . . . . . . . . . . . 16  |-  b  e. 
_V
3635brres 5138 . . . . . . . . . . . . . . 15  |-  ( a ( 1st  |`  ( _V  X.  _V ) ) b  <->  ( a 1st b  /\  a  e.  ( _V  X.  _V ) ) )
37 df-br 4200 . . . . . . . . . . . . . . 15  |-  ( a ( 1st  |`  ( _V  X.  _V ) ) b  <->  <. a ,  b
>.  e.  ( 1st  |`  ( _V  X.  _V ) ) )
38 ancom 438 . . . . . . . . . . . . . . 15  |-  ( ( a 1st b  /\  a  e.  ( _V  X.  _V ) )  <->  ( a  e.  ( _V  X.  _V )  /\  a 1st b
) )
3936, 37, 383bitr3ri 268 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ( _V 
X.  _V )  /\  a 1st b )  <->  <. a ,  b >.  e.  ( 1st  |`  ( _V  X.  _V ) ) )
4039anbi2i 676 . . . . . . . . . . . . 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 4922 . . . . . . . . . . . . . . . 16  |-  ( a  e.  ( _V  X.  _V )  <->  E. p E. q 
a  =  <. p ,  q >. )
4241anbi1i 677 . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . 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 438 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  =  <. p ,  q >.  /\  (
p  =  b  /\  q  =  x )
)  <->  ( ( p  =  b  /\  q  =  x )  /\  a  =  <. p ,  q
>. ) )
45 breq1 4202 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  <. p ,  q
>.  ->  ( a 1st b  <->  <. p ,  q
>. 1st b ) )
46 opeq1 3971 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( a  =  <. p ,  q
>.  ->  <. a ,  b
>.  =  <. <. p ,  q >. ,  b
>. )
4746breq1d 4209 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  =  <. p ,  q
>.  ->  ( <. a ,  b >. ( 2nd  o.  1st ) x  <->  <. <. p ,  q
>. ,  b >. ( 2nd  o.  1st )
x ) )
4845, 47anbi12d 692 . . . . . . . . . . . . . . . . . . . 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 2946 . . . . . . . . . . . . . . . . . . . . . . 23  |-  p  e. 
_V
50 vex 2946 . . . . . . . . . . . . . . . . . . . . . . 23  |-  q  e. 
_V
5149, 50, 35br1steq 25380 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
p ,  q >. 1st b  <->  b  =  p )
52 equcom 1692 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  =  p  <->  p  =  b )
5351, 52bitri 241 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
p ,  q >. 1st b  <->  p  =  b
)
54 opex 4414 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  <. <. p ,  q >. ,  b
>.  e.  _V
55 vex 2946 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  x  e. 
_V
5654, 55brco 5029 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <->  E. a ( <. <. p ,  q >. ,  b
>. 1st a  /\  a 2nd x ) )
57 opex 4414 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  <. p ,  q >.  e.  _V
5857, 35, 8br1steq 25380 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( <. <. p ,  q >. ,  b >. 1st a  <->  a  =  <. p ,  q
>. )
5958anbi1i 677 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
<. <. p ,  q
>. ,  b >. 1st a  /\  a 2nd x )  <->  ( a  =  <. p ,  q
>.  /\  a 2nd x
) )
6059exbii 1592 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E. a ( <. <. p ,  q >. ,  b
>. 1st a  /\  a 2nd x )  <->  E. a
( a  =  <. p ,  q >.  /\  a 2nd x ) )
6156, 60bitri 241 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <->  E. a ( a  = 
<. p ,  q >.  /\  a 2nd x ) )
62 breq1 4202 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( a  =  <. p ,  q
>.  ->  ( a 2nd x  <->  <. p ,  q
>. 2nd x ) )
6357, 62ceqsexv 2978 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( E. a ( a  = 
<. p ,  q >.  /\  a 2nd x )  <->  <. p ,  q >. 2nd x )
6449, 50, 55br2ndeq 25381 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
p ,  q >. 2nd x  <->  x  =  q
)
6563, 64bitri 241 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( E. a ( a  = 
<. p ,  q >.  /\  a 2nd x )  <-> 
x  =  q )
66 equcom 1692 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  q  <->  q  =  x )
6761, 65, 663bitri 263 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <. <. p ,  q >. ,  b >. ( 2nd  o.  1st ) x  <-> 
q  =  x )
6853, 67anbi12i 679 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
<. p ,  q >. 1st b  /\  <. <. p ,  q >. ,  b
>. ( 2nd  o.  1st ) x )  <->  ( p  =  b  /\  q  =  x ) )
6948, 68syl6bb 253 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  <. p ,  q
>.  ->  ( ( a 1st b  /\  <. a ,  b >. ( 2nd  o.  1st ) x )  <->  ( p  =  b  /\  q  =  x ) ) )
7069pm5.32i 619 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( a  =  <. p ,  q >.  /\  (
p  =  b  /\  q  =  x )
) )
71 df-3an 938 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )  <->  ( (
p  =  b  /\  q  =  x )  /\  a  =  <. p ,  q >. )
)
7244, 70, 713bitr4i 269 . . . . . . . . . . . . . . . . 17  |-  ( ( a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) )  <-> 
( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )
)
73722exbii 1593 . . . . . . . . . . . . . . . 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 1925 . . . . . . . . . . . . . . . 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 3971 . . . . . . . . . . . . . . . . . 18  |-  ( p  =  b  ->  <. p ,  q >.  =  <. b ,  q >. )
7675eqeq2d 2441 . . . . . . . . . . . . . . . . 17  |-  ( p  =  b  ->  (
a  =  <. p ,  q >.  <->  a  =  <. b ,  q >.
) )
77 opeq2 3972 . . . . . . . . . . . . . . . . . 18  |-  ( q  =  x  ->  <. b ,  q >.  =  <. b ,  x >. )
7877eqeq2d 2441 . . . . . . . . . . . . . . . . 17  |-  ( q  =  x  ->  (
a  =  <. b ,  q >.  <->  a  =  <. b ,  x >. ) )
7935, 55, 76, 78ceqsex2v 2980 . . . . . . . . . . . . . . . 16  |-  ( E. p E. q ( p  =  b  /\  q  =  x  /\  a  =  <. p ,  q >. )  <->  a  =  <. b ,  x >. )
8073, 74, 793bitr3ri 268 . . . . . . . . . . . . . . 15  |-  ( a  =  <. b ,  x >.  <-> 
( E. p E. q  a  =  <. p ,  q >.  /\  (
a 1st b  /\  <.
a ,  b >.
( 2nd  o.  1st ) x ) ) )
8142, 43, 803bitr4ri 270 . . . . . . . . . . . . . 14  |-  ( a  =  <. b ,  x >.  <-> 
( ( a  e.  ( _V  X.  _V )  /\  a 1st b
)  /\  <. a ,  b >. ( 2nd  o.  1st ) x ) )
82 ancom 438 . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . 13  |-  ( a  =  <. b ,  x >.  <-> 
( <. a ,  b
>. ( 2nd  o.  1st ) x  /\  (
a  e.  ( _V 
X.  _V )  /\  a 1st b ) ) )
8455brres 5138 . . . . . . . . . . . . 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 269 . . . . . . . . . . . 12  |-  ( a  =  <. b ,  x >.  <->  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
8685rexbii 2717 . . . . . . . . . . 11  |-  ( E. a  e.  A  a  =  <. b ,  x >.  <->  E. a  e.  A  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
8734, 86bitri 241 . . . . . . . . . 10  |-  ( <.
b ,  x >.  e.  A  <->  E. a  e.  A  <. a ,  b >.
( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x )
88 df-rex 2698 . . . . . . . . . 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 263 . . . . . . . . 9  |-  ( b A x  <->  E. a
( a  e.  A  /\  <. a ,  b
>. ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) x ) )
9089anbi2i 676 . . . . . . . 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 270 . . . . . . 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 1592 . . . . . 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 5195 . . . . . 6  |-  ( x  e.  ( A " B )  <->  E. b
( b  e.  B  /\  b A x ) )
9455elima2 5195 . . . . . . 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 4881 . . . . . . . . . 10  |-  ( p  e.  ( A  X.  B )  <->  E. a E. b ( p  = 
<. a ,  b >.  /\  ( a  e.  A  /\  b  e.  B
) ) )
9695anbi1i 677 . . . . . . . . 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 1925 . . . . . . . . 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 244 . . . . . . . 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 1592 . . . . . . 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 1759 . . . . . . . 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 1759 . . . . . . . 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 241 . . . . . . 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 263 . . . . . 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 270 . . . . 5  |-  ( x  e.  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <-> 
x  e.  ( A
" B ) )
105104eqriv 2427 . . . 4  |-  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  =  ( A " B )
106105eqeq2i 2440 . . 3  |-  ( C  =  ( ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) " ( A  X.  B ) )  <-> 
C  =  ( A
" B ) )
10716, 106bitri 241 . 2  |-  ( ( A  X.  B )Image ( ( 2nd  o.  1st )  |`  ( 1st  |`  ( _V  X.  _V ) ) ) C  <-> 
C  =  ( A
" B ) )
1082, 15, 1073bitri 263 1  |-  ( <. A ,  B >.Img 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   E.wrex 2693   _Vcvv 2943   <.cop 3804   class class class wbr 4199    X. cxp 4862    |` cres 4866   "cima 4867    o. ccom 4868   1stc1st 6333   2ndc2nd 6334  Imagecimage 25633  Cartccart 25634  Imgcimg 25635
This theorem is referenced by:  brapply  25728  dfrdg4  25740
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
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 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-eprel 4481  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-fo 5446  df-fv 5448  df-1st 6335  df-2nd 6336  df-symdif 25612  df-txp 25645  df-pprod 25646  df-image 25655  df-cart 25656  df-img 25657
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