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Theorem f1od2 5959
Description: Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
Hypotheses
Ref Expression
f1od2.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
f1od2.2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  e.  W )
f1od2.3  |-  ( (
ph  /\  z  e.  D )  ->  (
I  e.  X  /\  J  e.  Y )
)
f1od2.4  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( z  e.  D  /\  (
x  =  I  /\  y  =  J )
) ) )
Assertion
Ref Expression
f1od2  |-  ( ph  ->  F : ( A  X.  B ) -1-1-onto-> D )
Distinct variable groups:    x, y, z, A    x, B, y, z    z, C    x, D, y, z    x, I, y    x, J, y    ph, x, y, z
Allowed substitution hints:    C( x, y)    F( x, y, z)    I(
z)    J( z)    W( x, y, z)    X( x, y, z)    Y( x, y, z)

Proof of Theorem f1od2
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 f1od2.2 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  e.  W )
21ralrimivva 2451 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  B  C  e.  W )
3 f1od2.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
43fnmpt2 5931 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  e.  W  ->  F  Fn  ( A  X.  B
) )
52, 4syl 14 . 2  |-  ( ph  ->  F  Fn  ( A  X.  B ) )
6 f1od2.3 . . . . . 6  |-  ( (
ph  /\  z  e.  D )  ->  (
I  e.  X  /\  J  e.  Y )
)
7 opelxpi 4444 . . . . . 6  |-  ( ( I  e.  X  /\  J  e.  Y )  -> 
<. I ,  J >.  e.  ( X  X.  Y
) )
86, 7syl 14 . . . . 5  |-  ( (
ph  /\  z  e.  D )  ->  <. I ,  J >.  e.  ( X  X.  Y ) )
98ralrimiva 2442 . . . 4  |-  ( ph  ->  A. z  e.  D  <. I ,  J >.  e.  ( X  X.  Y
) )
10 eqid 2085 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  ( z  e.  D  |->  <.
I ,  J >. )
1110fnmpt 5107 . . . 4  |-  ( A. z  e.  D  <. I ,  J >.  e.  ( X  X.  Y )  ->  ( z  e.  D  |->  <. I ,  J >. )  Fn  D )
129, 11syl 14 . . 3  |-  ( ph  ->  ( z  e.  D  |-> 
<. I ,  J >. )  Fn  D )
13 elxp7 5900 . . . . . . . 8  |-  ( a  e.  ( A  X.  B )  <->  ( a  e.  ( _V  X.  _V )  /\  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a
)  e.  B ) ) )
1413anbi1i 446 . . . . . . 7  |-  ( ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  <->  ( (
a  e.  ( _V 
X.  _V )  /\  (
( 1st `  a
)  e.  A  /\  ( 2nd `  a )  e.  B ) )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C ) )
15 anass 393 . . . . . . . . 9  |-  ( ( ( a  e.  ( _V  X.  _V )  /\  ( ( 1st `  a
)  e.  A  /\  ( 2nd `  a )  e.  B ) )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )  <->  ( a  e.  ( _V  X.  _V )  /\  ( ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) ) )
16 f1od2.4 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( z  e.  D  /\  (
x  =  I  /\  y  =  J )
) ) )
1716sbcbidv 2886 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. ( 2nd `  a )  /  y ]. ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  [. ( 2nd `  a )  /  y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) ) ) )
1817sbcbidv 2886 . . . . . . . . . . 11  |-  ( ph  ->  ( [. ( 1st `  a )  /  x ]. [. ( 2nd `  a
)  /  y ]. ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  [. ( 1st `  a )  /  x ]. [. ( 2nd `  a
)  /  y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) ) ) )
19 sbcan 2870 . . . . . . . . . . . . . 14  |-  ( [. ( 2nd `  a )  /  y ]. (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  <->  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  /\  [. ( 2nd `  a
)  /  y ]. z  =  C )
)
20 sbcan 2870 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  <->  (
[. ( 2nd `  a
)  /  y ]. x  e.  A  /\  [. ( 2nd `  a
)  /  y ]. y  e.  B )
)
21 vex 2618 . . . . . . . . . . . . . . . . . . 19  |-  a  e. 
_V
22 2ndexg 5898 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  _V  ->  ( 2nd `  a )  e. 
_V )
2321, 22ax-mp 7 . . . . . . . . . . . . . . . . . 18  |-  ( 2nd `  a )  e.  _V
24 sbcg 2897 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. x  e.  A  <->  x  e.  A ) )
2523, 24ax-mp 7 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. x  e.  A  <->  x  e.  A
)
26 sbcel1v 2890 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. y  e.  B  <->  ( 2nd `  a
)  e.  B )
2725, 26anbi12i 448 . . . . . . . . . . . . . . . 16  |-  ( (
[. ( 2nd `  a
)  /  y ]. x  e.  A  /\  [. ( 2nd `  a
)  /  y ]. y  e.  B )  <->  ( x  e.  A  /\  ( 2nd `  a )  e.  B ) )
2820, 27bitri 182 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  <->  ( x  e.  A  /\  ( 2nd `  a )  e.  B ) )
29 sbceq2g 2942 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. z  =  C  <->  z  =  [_ ( 2nd `  a
)  /  y ]_ C ) )
3023, 29ax-mp 7 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. z  =  C  <->  z  =  [_ ( 2nd `  a )  /  y ]_ C
)
3128, 30anbi12i 448 . . . . . . . . . . . . . 14  |-  ( (
[. ( 2nd `  a
)  /  y ]. ( x  e.  A  /\  y  e.  B
)  /\  [. ( 2nd `  a )  /  y ]. z  =  C
)  <->  ( ( x  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 2nd `  a
)  /  y ]_ C ) )
3219, 31bitri 182 . . . . . . . . . . . . 13  |-  ( [. ( 2nd `  a )  /  y ]. (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  <->  ( (
x  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 2nd `  a )  /  y ]_ C ) )
3332sbcbii 2887 . . . . . . . . . . . 12  |-  ( [. ( 1st `  a )  /  x ]. [. ( 2nd `  a )  / 
y ]. ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  [. ( 1st `  a )  /  x ]. ( ( x  e.  A  /\  ( 2nd `  a )  e.  B
)  /\  z  =  [_ ( 2nd `  a
)  /  y ]_ C ) )
34 sbcan 2870 . . . . . . . . . . . 12  |-  ( [. ( 1st `  a )  /  x ]. (
( x  e.  A  /\  ( 2nd `  a
)  e.  B )  /\  z  =  [_ ( 2nd `  a )  /  y ]_ C
)  <->  ( [. ( 1st `  a )  /  x ]. ( x  e.  A  /\  ( 2nd `  a )  e.  B
)  /\  [. ( 1st `  a )  /  x ]. z  =  [_ ( 2nd `  a )  / 
y ]_ C ) )
35 sbcan 2870 . . . . . . . . . . . . . 14  |-  ( [. ( 1st `  a )  /  x ]. (
x  e.  A  /\  ( 2nd `  a )  e.  B )  <->  ( [. ( 1st `  a )  /  x ]. x  e.  A  /\  [. ( 1st `  a )  /  x ]. ( 2nd `  a
)  e.  B ) )
36 sbcel1v 2890 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. x  e.  A  <->  ( 1st `  a
)  e.  A )
37 1stexg 5897 . . . . . . . . . . . . . . . . 17  |-  ( a  e.  _V  ->  ( 1st `  a )  e. 
_V )
3821, 37ax-mp 7 . . . . . . . . . . . . . . . 16  |-  ( 1st `  a )  e.  _V
39 sbcg 2897 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  e.  B  <->  ( 2nd `  a )  e.  B
) )
4038, 39ax-mp 7 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. ( 2nd `  a )  e.  B  <->  ( 2nd `  a
)  e.  B )
4136, 40anbi12i 448 . . . . . . . . . . . . . 14  |-  ( (
[. ( 1st `  a
)  /  x ]. x  e.  A  /\  [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  e.  B )  <->  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B
) )
4235, 41bitri 182 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. (
x  e.  A  /\  ( 2nd `  a )  e.  B )  <->  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B
) )
43 sbceq2g 2942 . . . . . . . . . . . . . 14  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. z  =  [_ ( 2nd `  a )  /  y ]_ C  <->  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C ) )
4438, 43ax-mp 7 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. z  =  [_ ( 2nd `  a
)  /  y ]_ C 
<->  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  /  y ]_ C )
4542, 44anbi12i 448 . . . . . . . . . . . 12  |-  ( (
[. ( 1st `  a
)  /  x ]. ( x  e.  A  /\  ( 2nd `  a
)  e.  B )  /\  [. ( 1st `  a )  /  x ]. z  =  [_ ( 2nd `  a )  / 
y ]_ C )  <->  ( (
( 1st `  a
)  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) )
4633, 34, 453bitri 204 . . . . . . . . . . 11  |-  ( [. ( 1st `  a )  /  x ]. [. ( 2nd `  a )  / 
y ]. ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
( 1st `  a
)  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) )
47 sbcan 2870 . . . . . . . . . . . . . 14  |-  ( [. ( 2nd `  a )  /  y ]. (
z  e.  D  /\  ( x  =  I  /\  y  =  J
) )  <->  ( [. ( 2nd `  a )  /  y ]. z  e.  D  /\  [. ( 2nd `  a )  / 
y ]. ( x  =  I  /\  y  =  J ) ) )
48 sbcg 2897 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. z  e.  D  <->  z  e.  D ) )
4923, 48ax-mp 7 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. z  e.  D  <->  z  e.  D
)
50 sbcan 2870 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  =  I  /\  y  =  J )  <->  (
[. ( 2nd `  a
)  /  y ]. x  =  I  /\  [. ( 2nd `  a
)  /  y ]. y  =  J )
)
51 sbcg 2897 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. x  =  I  <->  x  =  I ) )
5223, 51ax-mp 7 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. x  =  I  <->  x  =  I
)
53 sbceq1g 2940 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. y  =  J  <->  [_ ( 2nd `  a )  /  y ]_ y  =  J
) )
5423, 53ax-mp 7 . . . . . . . . . . . . . . . . . 18  |-  ( [. ( 2nd `  a )  /  y ]. y  =  J  <->  [_ ( 2nd `  a
)  /  y ]_ y  =  J )
55 csbvarg 2947 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  a )  e.  _V  ->  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a ) )
5623, 55ax-mp 7 . . . . . . . . . . . . . . . . . . 19  |-  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a )
5756eqeq1i 2092 . . . . . . . . . . . . . . . . . 18  |-  ( [_ ( 2nd `  a )  /  y ]_ y  =  J  <->  ( 2nd `  a
)  =  J )
5854, 57bitri 182 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. y  =  J  <->  ( 2nd `  a
)  =  J )
5952, 58anbi12i 448 . . . . . . . . . . . . . . . 16  |-  ( (
[. ( 2nd `  a
)  /  y ]. x  =  I  /\  [. ( 2nd `  a
)  /  y ]. y  =  J )  <->  ( x  =  I  /\  ( 2nd `  a )  =  J ) )
6050, 59bitri 182 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. (
x  =  I  /\  y  =  J )  <->  ( x  =  I  /\  ( 2nd `  a )  =  J ) )
6149, 60anbi12i 448 . . . . . . . . . . . . . 14  |-  ( (
[. ( 2nd `  a
)  /  y ]. z  e.  D  /\  [. ( 2nd `  a
)  /  y ]. ( x  =  I  /\  y  =  J
) )  <->  ( z  e.  D  /\  (
x  =  I  /\  ( 2nd `  a )  =  J ) ) )
6247, 61bitri 182 . . . . . . . . . . . . 13  |-  ( [. ( 2nd `  a )  /  y ]. (
z  e.  D  /\  ( x  =  I  /\  y  =  J
) )  <->  ( z  e.  D  /\  (
x  =  I  /\  ( 2nd `  a )  =  J ) ) )
6362sbcbii 2887 . . . . . . . . . . . 12  |-  ( [. ( 1st `  a )  /  x ]. [. ( 2nd `  a )  / 
y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) )  <->  [. ( 1st `  a )  /  x ]. ( z  e.  D  /\  ( x  =  I  /\  ( 2nd `  a
)  =  J ) ) )
64 sbcan 2870 . . . . . . . . . . . 12  |-  ( [. ( 1st `  a )  /  x ]. (
z  e.  D  /\  ( x  =  I  /\  ( 2nd `  a
)  =  J ) )  <->  ( [. ( 1st `  a )  /  x ]. z  e.  D  /\  [. ( 1st `  a
)  /  x ]. ( x  =  I  /\  ( 2nd `  a
)  =  J ) ) )
65 sbcg 2897 . . . . . . . . . . . . . 14  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. z  e.  D  <->  z  e.  D ) )
6638, 65ax-mp 7 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. z  e.  D  <->  z  e.  D
)
67 sbcan 2870 . . . . . . . . . . . . . 14  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( [. ( 1st `  a )  /  x ]. x  =  I  /\  [. ( 1st `  a )  /  x ]. ( 2nd `  a
)  =  J ) )
68 sbceq1g 2940 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. x  =  I  <->  [_ ( 1st `  a )  /  x ]_ x  =  I
) )
6938, 68ax-mp 7 . . . . . . . . . . . . . . . 16  |-  ( [. ( 1st `  a )  /  x ]. x  =  I  <->  [_ ( 1st `  a
)  /  x ]_ x  =  I )
70 csbvarg 2947 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  a )  e.  _V  ->  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a ) )
7138, 70ax-mp 7 . . . . . . . . . . . . . . . . 17  |-  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a )
7271eqeq1i 2092 . . . . . . . . . . . . . . . 16  |-  ( [_ ( 1st `  a )  /  x ]_ x  =  I  <->  ( 1st `  a
)  =  I )
7369, 72bitri 182 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. x  =  I  <->  ( 1st `  a
)  =  I )
74 sbcg 2897 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  =  J  <->  ( 2nd `  a )  =  J ) )
7538, 74ax-mp 7 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. ( 2nd `  a )  =  J  <->  ( 2nd `  a
)  =  J )
7673, 75anbi12i 448 . . . . . . . . . . . . . 14  |-  ( (
[. ( 1st `  a
)  /  x ]. x  =  I  /\  [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7767, 76bitri 182 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7866, 77anbi12i 448 . . . . . . . . . . . 12  |-  ( (
[. ( 1st `  a
)  /  x ]. z  e.  D  /\  [. ( 1st `  a
)  /  x ]. ( x  =  I  /\  ( 2nd `  a
)  =  J ) )  <->  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )
7963, 64, 783bitri 204 . . . . . . . . . . 11  |-  ( [. ( 1st `  a )  /  x ]. [. ( 2nd `  a )  / 
y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) )
8018, 46, 793bitr3g 220 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) ) )
8180anbi2d 452 . . . . . . . . 9  |-  ( ph  ->  ( ( a  e.  ( _V  X.  _V )  /\  ( ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) )  <->  ( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) ) ) )
8215, 81syl5bb 190 . . . . . . . 8  |-  ( ph  ->  ( ( ( a  e.  ( _V  X.  _V )  /\  (
( 1st `  a
)  e.  A  /\  ( 2nd `  a )  e.  B ) )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )  <->  ( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) ) ) )
83 xpss 4516 . . . . . . . . . . . 12  |-  ( X  X.  Y )  C_  ( _V  X.  _V )
84 simprr 499 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  =  <. I ,  J >. )
858adantrr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  <. I ,  J >.  e.  ( X  X.  Y ) )
8684, 85eqeltrd 2161 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( X  X.  Y
) )
8783, 86sseldi 3012 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( _V  X.  _V ) )
8887ex 113 . . . . . . . . . 10  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  ->  a  e.  ( _V  X.  _V )
) )
8988pm4.71rd 386 . . . . . . . . 9  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  <-> 
( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  a  =  <. I ,  J >. )
) ) )
90 eqop 5906 . . . . . . . . . . 11  |-  ( a  e.  ( _V  X.  _V )  ->  ( a  =  <. I ,  J >.  <-> 
( ( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) )
9190anbi2d 452 . . . . . . . . . 10  |-  ( a  e.  ( _V  X.  _V )  ->  ( ( z  e.  D  /\  a  =  <. I ,  J >. )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) ) )
9291pm5.32i 442 . . . . . . . . 9  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
z  e.  D  /\  a  =  <. I ,  J >. ) )  <->  ( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) ) )
9389, 92syl6rbb 195 . . . . . . . 8  |-  ( ph  ->  ( ( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )  <-> 
( z  e.  D  /\  a  =  <. I ,  J >. )
) )
9482, 93bitrd 186 . . . . . . 7  |-  ( ph  ->  ( ( ( a  e.  ( _V  X.  _V )  /\  (
( 1st `  a
)  e.  A  /\  ( 2nd `  a )  e.  B ) )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )  <->  ( z  e.  D  /\  a  =  <. I ,  J >. ) ) )
9514, 94syl5bb 190 . . . . . 6  |-  ( ph  ->  ( ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  <->  ( z  e.  D  /\  a  =  <. I ,  J >. ) ) )
9695opabbidv 3881 . . . . 5  |-  ( ph  ->  { <. z ,  a
>.  |  ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) }  =  { <. z ,  a
>.  |  ( z  e.  D  /\  a  =  <. I ,  J >. ) } )
97 df-mpt2 5620 . . . . . . . 8  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
983, 97eqtri 2105 . . . . . . 7  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
9998cnveqi 4581 . . . . . 6  |-  `' F  =  `' { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
100 nfv 1464 . . . . . . . 8  |-  F/ x  a  e.  ( A  X.  B )
101 nfcsb1v 2952 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
102101nfeq2 2236 . . . . . . . 8  |-  F/ x  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
103100, 102nfan 1500 . . . . . . 7  |-  F/ x
( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
104 nfv 1464 . . . . . . . 8  |-  F/ y  a  e.  ( A  X.  B )
105 nfcv 2225 . . . . . . . . . 10  |-  F/_ y
( 1st `  a
)
106 nfcsb1v 2952 . . . . . . . . . 10  |-  F/_ y [_ ( 2nd `  a
)  /  y ]_ C
107105, 106nfcsb 2954 . . . . . . . . 9  |-  F/_ y [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
108107nfeq2 2236 . . . . . . . 8  |-  F/ y  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  /  y ]_ C
109104, 108nfan 1500 . . . . . . 7  |-  F/ y ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
110 eleq1 2147 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
111 opelxp 4442 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
112110, 111syl6bb 194 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
113 csbopeq1a 5917 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C  =  C )
114113eqeq2d 2096 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( z  = 
[_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C 
<->  z  =  C ) )
115112, 114anbi12d 457 . . . . . . 7  |-  ( a  =  <. x ,  y
>.  ->  ( ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) ) )
116 xpss 4516 . . . . . . . . 9  |-  ( A  X.  B )  C_  ( _V  X.  _V )
117116sseli 3010 . . . . . . . 8  |-  ( a  e.  ( A  X.  B )  ->  a  e.  ( _V  X.  _V ) )
118117adantr 270 . . . . . . 7  |-  ( ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  ->  a  e.  ( _V  X.  _V ) )
119103, 109, 115, 118cnvoprab 5958 . . . . . 6  |-  `' { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }  =  { <. z ,  a
>.  |  ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) }
12099, 119eqtri 2105 . . . . 5  |-  `' F  =  { <. z ,  a
>.  |  ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) }
121 df-mpt 3878 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  { <. z ,  a >.  |  ( z  e.  D  /\  a  = 
<. I ,  J >. ) }
12296, 120, 1213eqtr4g 2142 . . . 4  |-  ( ph  ->  `' F  =  (
z  e.  D  |->  <.
I ,  J >. ) )
123122fneq1d 5071 . . 3  |-  ( ph  ->  ( `' F  Fn  D 
<->  ( z  e.  D  |-> 
<. I ,  J >. )  Fn  D ) )
12412, 123mpbird 165 . 2  |-  ( ph  ->  `' F  Fn  D
)
125 dff1o4 5226 . 2  |-  ( F : ( A  X.  B ) -1-1-onto-> D  <->  ( F  Fn  ( A  X.  B
)  /\  `' F  Fn  D ) )
1265, 124, 125sylanbrc 408 1  |-  ( ph  ->  F : ( A  X.  B ) -1-1-onto-> D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   A.wral 2355   _Vcvv 2615   [.wsbc 2829   [_csb 2922   <.cop 3434   {copab 3875    |-> cmpt 3876    X. cxp 4411   `'ccnv 4412    Fn wfn 4978   -1-1-onto->wf1o 4982   ` cfv 4983   {coprab 5616    |-> cmpt2 5617   1stc1st 5868   2ndc2nd 5869
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871
This theorem is referenced by:  oddpwdc  11058
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