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Theorem f1od2 27704
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 2878 . . 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 6867 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  e.  W  ->  F  Fn  ( A  X.  B
) )
52, 4syl 16 . 2  |-  ( ph  ->  F  Fn  ( A  X.  B ) )
6 f1od2.3 . . . . . 6  |-  ( (
ph  /\  z  e.  D )  ->  (
I  e.  X  /\  J  e.  Y )
)
7 opelxpi 5040 . . . . . 6  |-  ( ( I  e.  X  /\  J  e.  Y )  -> 
<. I ,  J >.  e.  ( X  X.  Y
) )
86, 7syl 16 . . . . 5  |-  ( (
ph  /\  z  e.  D )  ->  <. I ,  J >.  e.  ( X  X.  Y ) )
98ralrimiva 2871 . . . 4  |-  ( ph  ->  A. z  e.  D  <. I ,  J >.  e.  ( X  X.  Y
) )
10 eqid 2457 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  ( z  e.  D  |->  <.
I ,  J >. )
1110fnmpt 5713 . . . 4  |-  ( A. z  e.  D  <. I ,  J >.  e.  ( X  X.  Y )  ->  ( z  e.  D  |->  <. I ,  J >. )  Fn  D )
129, 11syl 16 . . 3  |-  ( ph  ->  ( z  e.  D  |-> 
<. I ,  J >. )  Fn  D )
13 elxp7 6832 . . . . . . . 8  |-  ( a  e.  ( A  X.  B )  <->  ( a  e.  ( _V  X.  _V )  /\  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a
)  e.  B ) ) )
1413anbi1i 695 . . . . . . 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 649 . . . . . . . . 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 3386 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. ( 2nd `  a )  /  y ]. ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  [. ( 2nd `  a )  /  y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) ) ) )
1817sbcbidv 3386 . . . . . . . . . . 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 3370 . . . . . . . . . . . . . 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 3370 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  <->  (
[. ( 2nd `  a
)  /  y ]. x  e.  A  /\  [. ( 2nd `  a
)  /  y ]. y  e.  B )
)
21 fvex 5882 . . . . . . . . . . . . . . . . . 18  |-  ( 2nd `  a )  e.  _V
22 sbcg 3399 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. x  e.  A  <->  x  e.  A ) )
2321, 22ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. x  e.  A  <->  x  e.  A
)
24 sbcel1v 3391 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. y  e.  B  <->  ( 2nd `  a
)  e.  B )
2523, 24anbi12i 697 . . . . . . . . . . . . . . . 16  |-  ( (
[. ( 2nd `  a
)  /  y ]. x  e.  A  /\  [. ( 2nd `  a
)  /  y ]. y  e.  B )  <->  ( x  e.  A  /\  ( 2nd `  a )  e.  B ) )
2620, 25bitri 249 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  <->  ( x  e.  A  /\  ( 2nd `  a )  e.  B ) )
27 sbceq2g 3841 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. z  =  C  <->  z  =  [_ ( 2nd `  a
)  /  y ]_ C ) )
2821, 27ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. z  =  C  <->  z  =  [_ ( 2nd `  a )  /  y ]_ C
)
2926, 28anbi12i 697 . . . . . . . . . . . . . 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 ) )
3019, 29bitri 249 . . . . . . . . . . . . 13  |-  ( [. ( 2nd `  a )  /  y ]. (
( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  <->  ( (
x  e.  A  /\  ( 2nd `  a )  e.  B )  /\  z  =  [_ ( 2nd `  a )  /  y ]_ C ) )
3130sbcbii 3387 . . . . . . . . . . . 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 ) )
32 sbcan 3370 . . . . . . . . . . . 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 ) )
33 sbcan 3370 . . . . . . . . . . . . . 14  |-  ( [. ( 1st `  a )  /  x ]. (
x  e.  A  /\  ( 2nd `  a )  e.  B )  <->  ( [. ( 1st `  a )  /  x ]. x  e.  A  /\  [. ( 1st `  a )  /  x ]. ( 2nd `  a
)  e.  B ) )
34 sbcel1v 3391 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. x  e.  A  <->  ( 1st `  a
)  e.  A )
35 fvex 5882 . . . . . . . . . . . . . . . 16  |-  ( 1st `  a )  e.  _V
36 sbcg 3399 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  e.  B  <->  ( 2nd `  a )  e.  B
) )
3735, 36ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. ( 2nd `  a )  e.  B  <->  ( 2nd `  a
)  e.  B )
3834, 37anbi12i 697 . . . . . . . . . . . . . 14  |-  ( (
[. ( 1st `  a
)  /  x ]. x  e.  A  /\  [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  e.  B )  <->  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B
) )
3933, 38bitri 249 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. (
x  e.  A  /\  ( 2nd `  a )  e.  B )  <->  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B
) )
40 sbceq2g 3841 . . . . . . . . . . . . . 14  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. z  =  [_ ( 2nd `  a )  /  y ]_ C  <->  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C ) )
4135, 40ax-mp 5 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. z  =  [_ ( 2nd `  a
)  /  y ]_ C 
<->  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  /  y ]_ C )
4239, 41anbi12i 697 . . . . . . . . . . . 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 ) )
4331, 32, 423bitri 271 . . . . . . . . . . 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 ) )
44 sbcan 3370 . . . . . . . . . . . . . 14  |-  ( [. ( 2nd `  a )  /  y ]. (
z  e.  D  /\  ( x  =  I  /\  y  =  J
) )  <->  ( [. ( 2nd `  a )  /  y ]. z  e.  D  /\  [. ( 2nd `  a )  / 
y ]. ( x  =  I  /\  y  =  J ) ) )
45 sbcg 3399 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. z  e.  D  <->  z  e.  D ) )
4621, 45ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. z  e.  D  <->  z  e.  D
)
47 sbcan 3370 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  =  I  /\  y  =  J )  <->  (
[. ( 2nd `  a
)  /  y ]. x  =  I  /\  [. ( 2nd `  a
)  /  y ]. y  =  J )
)
48 sbcg 3399 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. x  =  I  <->  x  =  I ) )
4921, 48ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. x  =  I  <->  x  =  I
)
50 sbceq1g 3838 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  a )  e.  _V  ->  ( [. ( 2nd `  a
)  /  y ]. y  =  J  <->  [_ ( 2nd `  a )  /  y ]_ y  =  J
) )
5121, 50ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  ( [. ( 2nd `  a )  /  y ]. y  =  J  <->  [_ ( 2nd `  a
)  /  y ]_ y  =  J )
52 csbvarg 3855 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  a )  e.  _V  ->  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a ) )
5321, 52ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a )
5453eqeq1i 2464 . . . . . . . . . . . . . . . . . 18  |-  ( [_ ( 2nd `  a )  /  y ]_ y  =  J  <->  ( 2nd `  a
)  =  J )
5551, 54bitri 249 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. y  =  J  <->  ( 2nd `  a
)  =  J )
5649, 55anbi12i 697 . . . . . . . . . . . . . . . 16  |-  ( (
[. ( 2nd `  a
)  /  y ]. x  =  I  /\  [. ( 2nd `  a
)  /  y ]. y  =  J )  <->  ( x  =  I  /\  ( 2nd `  a )  =  J ) )
5747, 56bitri 249 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. (
x  =  I  /\  y  =  J )  <->  ( x  =  I  /\  ( 2nd `  a )  =  J ) )
5846, 57anbi12i 697 . . . . . . . . . . . . . 14  |-  ( (
[. ( 2nd `  a
)  /  y ]. z  e.  D  /\  [. ( 2nd `  a
)  /  y ]. ( x  =  I  /\  y  =  J
) )  <->  ( z  e.  D  /\  (
x  =  I  /\  ( 2nd `  a )  =  J ) ) )
5944, 58bitri 249 . . . . . . . . . . . . 13  |-  ( [. ( 2nd `  a )  /  y ]. (
z  e.  D  /\  ( x  =  I  /\  y  =  J
) )  <->  ( z  e.  D  /\  (
x  =  I  /\  ( 2nd `  a )  =  J ) ) )
6059sbcbii 3387 . . . . . . . . . . . 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 ) ) )
61 sbcan 3370 . . . . . . . . . . . 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 ) ) )
62 sbcg 3399 . . . . . . . . . . . . . 14  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. z  e.  D  <->  z  e.  D ) )
6335, 62ax-mp 5 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. z  e.  D  <->  z  e.  D
)
64 sbcan 3370 . . . . . . . . . . . . . 14  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( [. ( 1st `  a )  /  x ]. x  =  I  /\  [. ( 1st `  a )  /  x ]. ( 2nd `  a
)  =  J ) )
65 sbceq1g 3838 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. x  =  I  <->  [_ ( 1st `  a )  /  x ]_ x  =  I
) )
6635, 65ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( [. ( 1st `  a )  /  x ]. x  =  I  <->  [_ ( 1st `  a
)  /  x ]_ x  =  I )
67 csbvarg 3855 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  a )  e.  _V  ->  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a ) )
6835, 67ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a )
6968eqeq1i 2464 . . . . . . . . . . . . . . . 16  |-  ( [_ ( 1st `  a )  /  x ]_ x  =  I  <->  ( 1st `  a
)  =  I )
7066, 69bitri 249 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. x  =  I  <->  ( 1st `  a
)  =  I )
71 sbcg 3399 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  =  J  <->  ( 2nd `  a )  =  J ) )
7235, 71ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. ( 2nd `  a )  =  J  <->  ( 2nd `  a
)  =  J )
7370, 72anbi12i 697 . . . . . . . . . . . . . 14  |-  ( (
[. ( 1st `  a
)  /  x ]. x  =  I  /\  [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7464, 73bitri 249 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7563, 74anbi12i 697 . . . . . . . . . . . 12  |-  ( (
[. ( 1st `  a
)  /  x ]. z  e.  D  /\  [. ( 1st `  a
)  /  x ]. ( x  =  I  /\  ( 2nd `  a
)  =  J ) )  <->  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )
7660, 61, 753bitri 271 . . . . . . . . . . 11  |-  ( [. ( 1st `  a )  /  x ]. [. ( 2nd `  a )  / 
y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) )
7718, 43, 763bitr3g 287 . . . . . . . . . 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 ) ) ) )
7877anbi2d 703 . . . . . . . . 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 ) ) ) ) )
7915, 78syl5bb 257 . . . . . . . 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 ) ) ) ) )
80 xpss 5118 . . . . . . . . . . . 12  |-  ( X  X.  Y )  C_  ( _V  X.  _V )
81 simprr 757 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  =  <. I ,  J >. )
828adantrr 716 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  <. I ,  J >.  e.  ( X  X.  Y ) )
8381, 82eqeltrd 2545 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( X  X.  Y
) )
8480, 83sseldi 3497 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( _V  X.  _V ) )
8584ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  ->  a  e.  ( _V  X.  _V )
) )
8685pm4.71rd 635 . . . . . . . . 9  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  <-> 
( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  a  =  <. I ,  J >. )
) ) )
87 eqop 6839 . . . . . . . . . . 11  |-  ( a  e.  ( _V  X.  _V )  ->  ( a  =  <. I ,  J >.  <-> 
( ( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) )
8887anbi2d 703 . . . . . . . . . 10  |-  ( a  e.  ( _V  X.  _V )  ->  ( ( z  e.  D  /\  a  =  <. I ,  J >. )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) ) )
8988pm5.32i 637 . . . . . . . . 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 ) ) ) )
9086, 89syl6rbb 262 . . . . . . . 8  |-  ( ph  ->  ( ( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )  <-> 
( z  e.  D  /\  a  =  <. I ,  J >. )
) )
9179, 90bitrd 253 . . . . . . 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 >. ) ) )
9214, 91syl5bb 257 . . . . . 6  |-  ( ph  ->  ( ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  <->  ( z  e.  D  /\  a  =  <. I ,  J >. ) ) )
9392opabbidv 4520 . . . . 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 >. ) } )
94 df-mpt2 6301 . . . . . . . 8  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
953, 94eqtri 2486 . . . . . . 7  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
9695cnveqi 5187 . . . . . 6  |-  `' F  =  `' { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
97 nfv 1708 . . . . . . . 8  |-  F/ x  a  e.  ( A  X.  B )
98 nfcsb1v 3446 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
9998nfeq2 2636 . . . . . . . 8  |-  F/ x  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
10097, 99nfan 1929 . . . . . . 7  |-  F/ x
( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
101 nfv 1708 . . . . . . . 8  |-  F/ y  a  e.  ( A  X.  B )
102 nfcv 2619 . . . . . . . . . 10  |-  F/_ y
( 1st `  a
)
103 nfcsb1v 3446 . . . . . . . . . 10  |-  F/_ y [_ ( 2nd `  a
)  /  y ]_ C
104102, 103nfcsb 3448 . . . . . . . . 9  |-  F/_ y [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
105104nfeq2 2636 . . . . . . . 8  |-  F/ y  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  /  y ]_ C
106101, 105nfan 1929 . . . . . . 7  |-  F/ y ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
107 eleq1 2529 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
108 opelxp 5038 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
109107, 108syl6bb 261 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
110 csbopeq1a 6852 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C  =  C )
111110eqeq2d 2471 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( z  = 
[_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C 
<->  z  =  C ) )
112109, 111anbi12d 710 . . . . . . 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
) ) )
113 xpss 5118 . . . . . . . . 9  |-  ( A  X.  B )  C_  ( _V  X.  _V )
114113sseli 3495 . . . . . . . 8  |-  ( a  e.  ( A  X.  B )  ->  a  e.  ( _V  X.  _V ) )
115114adantr 465 . . . . . . 7  |-  ( ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  ->  a  e.  ( _V  X.  _V ) )
116100, 106, 112, 115cnvoprab 27703 . . . . . 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 ) }
11796, 116eqtri 2486 . . . . 5  |-  `' F  =  { <. z ,  a
>.  |  ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) }
118 df-mpt 4517 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  { <. z ,  a >.  |  ( z  e.  D  /\  a  = 
<. I ,  J >. ) }
11993, 117, 1183eqtr4g 2523 . . . 4  |-  ( ph  ->  `' F  =  (
z  e.  D  |->  <.
I ,  J >. ) )
120119fneq1d 5677 . . 3  |-  ( ph  ->  ( `' F  Fn  D 
<->  ( z  e.  D  |-> 
<. I ,  J >. )  Fn  D ) )
12112, 120mpbird 232 . 2  |-  ( ph  ->  `' F  Fn  D
)
122 dff1o4 5830 . 2  |-  ( F : ( A  X.  B ) -1-1-onto-> D  <->  ( F  Fn  ( A  X.  B
)  /\  `' F  Fn  D ) )
1235, 121, 122sylanbrc 664 1  |-  ( ph  ->  F : ( A  X.  B ) -1-1-onto-> D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327   [_csb 3430   <.cop 4038   {copab 4514    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007    Fn wfn 5589   -1-1-onto->wf1o 5593   ` cfv 5594   {coprab 6297    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800
This theorem is referenced by:  oddpwdc  28490
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