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Theorem f1od2 28384
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 2814 . . 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 6880 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  e.  W  ->  F  Fn  ( A  X.  B
) )
52, 4syl 17 . 2  |-  ( ph  ->  F  Fn  ( A  X.  B ) )
6 f1od2.3 . . . . . 6  |-  ( (
ph  /\  z  e.  D )  ->  (
I  e.  X  /\  J  e.  Y )
)
7 opelxpi 4871 . . . . . 6  |-  ( ( I  e.  X  /\  J  e.  Y )  -> 
<. I ,  J >.  e.  ( X  X.  Y
) )
86, 7syl 17 . . . . 5  |-  ( (
ph  /\  z  e.  D )  ->  <. I ,  J >.  e.  ( X  X.  Y ) )
98ralrimiva 2809 . . . 4  |-  ( ph  ->  A. z  e.  D  <. I ,  J >.  e.  ( X  X.  Y
) )
10 eqid 2471 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  ( z  e.  D  |->  <.
I ,  J >. )
1110fnmpt 5714 . . . 4  |-  ( A. z  e.  D  <. I ,  J >.  e.  ( X  X.  Y )  ->  ( z  e.  D  |->  <. I ,  J >. )  Fn  D )
129, 11syl 17 . . 3  |-  ( ph  ->  ( z  e.  D  |-> 
<. I ,  J >. )  Fn  D )
13 elxp7 6845 . . . . . . . 8  |-  ( a  e.  ( A  X.  B )  <->  ( a  e.  ( _V  X.  _V )  /\  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a
)  e.  B ) ) )
1413anbi1i 709 . . . . . . 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 661 . . . . . . . . 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 3310 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. ( 2nd `  a )  /  y ]. ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  [. ( 2nd `  a )  /  y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) ) ) )
1817sbcbidv 3310 . . . . . . . . . . 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 3298 . . . . . . . . . . . . . 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 3298 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  <->  (
[. ( 2nd `  a
)  /  y ]. x  e.  A  /\  [. ( 2nd `  a
)  /  y ]. y  e.  B )
)
21 fvex 5889 . . . . . . . . . . . . . . . . . 18  |-  ( 2nd `  a )  e.  _V
22 sbcg 3321 . . . . . . . . . . . . . . . . . 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 3314 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. y  e.  B  <->  ( 2nd `  a
)  e.  B )
2523, 24anbi12i 711 . . . . . . . . . . . . . . . 16  |-  ( (
[. ( 2nd `  a
)  /  y ]. x  e.  A  /\  [. ( 2nd `  a
)  /  y ]. y  e.  B )  <->  ( x  e.  A  /\  ( 2nd `  a )  e.  B ) )
2620, 25bitri 257 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  <->  ( x  e.  A  /\  ( 2nd `  a )  e.  B ) )
27 sbceq2g 3783 . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . 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 3311 . . . . . . . . . . . 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 3298 . . . . . . . . . . . 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 3298 . . . . . . . . . . . . . 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 3314 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. x  e.  A  <->  ( 1st `  a
)  e.  A )
35 fvex 5889 . . . . . . . . . . . . . . . 16  |-  ( 1st `  a )  e.  _V
36 sbcg 3321 . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . 14  |-  ( (
[. ( 1st `  a
)  /  x ]. x  e.  A  /\  [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  e.  B )  <->  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B
) )
3933, 38bitri 257 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. (
x  e.  A  /\  ( 2nd `  a )  e.  B )  <->  ( ( 1st `  a )  e.  A  /\  ( 2nd `  a )  e.  B
) )
40 sbceq2g 3783 . . . . . . . . . . . . . 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 711 . . . . . . . . . . . 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 279 . . . . . . . . . . 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 3298 . . . . . . . . . . . . . 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 3321 . . . . . . . . . . . . . . . 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 3298 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  =  I  /\  y  =  J )  <->  (
[. ( 2nd `  a
)  /  y ]. x  =  I  /\  [. ( 2nd `  a
)  /  y ]. y  =  J )
)
48 sbcg 3321 . . . . . . . . . . . . . . . . . 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 3781 . . . . . . . . . . . . . . . . . . 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 3796 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  a )  e.  _V  ->  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a ) )
5321, 52ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a )
5453eqeq1i 2476 . . . . . . . . . . . . . . . . . 18  |-  ( [_ ( 2nd `  a )  /  y ]_ y  =  J  <->  ( 2nd `  a
)  =  J )
5551, 54bitri 257 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 2nd `  a )  /  y ]. y  =  J  <->  ( 2nd `  a
)  =  J )
5649, 55anbi12i 711 . . . . . . . . . . . . . . . 16  |-  ( (
[. ( 2nd `  a
)  /  y ]. x  =  I  /\  [. ( 2nd `  a
)  /  y ]. y  =  J )  <->  ( x  =  I  /\  ( 2nd `  a )  =  J ) )
5747, 56bitri 257 . . . . . . . . . . . . . . 15  |-  ( [. ( 2nd `  a )  /  y ]. (
x  =  I  /\  y  =  J )  <->  ( x  =  I  /\  ( 2nd `  a )  =  J ) )
5846, 57anbi12i 711 . . . . . . . . . . . . . 14  |-  ( (
[. ( 2nd `  a
)  /  y ]. z  e.  D  /\  [. ( 2nd `  a
)  /  y ]. ( x  =  I  /\  y  =  J
) )  <->  ( z  e.  D  /\  (
x  =  I  /\  ( 2nd `  a )  =  J ) ) )
5944, 58bitri 257 . . . . . . . . . . . . 13  |-  ( [. ( 2nd `  a )  /  y ]. (
z  e.  D  /\  ( x  =  I  /\  y  =  J
) )  <->  ( z  e.  D  /\  (
x  =  I  /\  ( 2nd `  a )  =  J ) ) )
6059sbcbii 3311 . . . . . . . . . . . 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 3298 . . . . . . . . . . . 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 3321 . . . . . . . . . . . . . 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 3298 . . . . . . . . . . . . . 14  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( [. ( 1st `  a )  /  x ]. x  =  I  /\  [. ( 1st `  a )  /  x ]. ( 2nd `  a
)  =  J ) )
65 sbceq1g 3781 . . . . . . . . . . . . . . . . 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 3796 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  a )  e.  _V  ->  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a ) )
6835, 67ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a )
6968eqeq1i 2476 . . . . . . . . . . . . . . . 16  |-  ( [_ ( 1st `  a )  /  x ]_ x  =  I  <->  ( 1st `  a
)  =  I )
7066, 69bitri 257 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. x  =  I  <->  ( 1st `  a
)  =  I )
71 sbcg 3321 . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . 14  |-  ( (
[. ( 1st `  a
)  /  x ]. x  =  I  /\  [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7464, 73bitri 257 . . . . . . . . . . . . 13  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7563, 74anbi12i 711 . . . . . . . . . . . 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 279 . . . . . . . . . . 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 295 . . . . . . . . . 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 718 . . . . . . . . 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 265 . . . . . . . 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 4946 . . . . . . . . . . . 12  |-  ( X  X.  Y )  C_  ( _V  X.  _V )
81 simprr 774 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  =  <. I ,  J >. )
828adantrr 731 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  <. I ,  J >.  e.  ( X  X.  Y ) )
8381, 82eqeltrd 2549 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( X  X.  Y
) )
8480, 83sseldi 3416 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( _V  X.  _V ) )
8584ex 441 . . . . . . . . . 10  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  ->  a  e.  ( _V  X.  _V )
) )
8685pm4.71rd 647 . . . . . . . . 9  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  <-> 
( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  a  =  <. I ,  J >. )
) ) )
87 eqop 6852 . . . . . . . . . . 11  |-  ( a  e.  ( _V  X.  _V )  ->  ( a  =  <. I ,  J >.  <-> 
( ( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) )
8887anbi2d 718 . . . . . . . . . 10  |-  ( a  e.  ( _V  X.  _V )  ->  ( ( z  e.  D  /\  a  =  <. I ,  J >. )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) ) )
8988pm5.32i 649 . . . . . . . . 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 270 . . . . . . . 8  |-  ( ph  ->  ( ( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )  <-> 
( z  e.  D  /\  a  =  <. I ,  J >. )
) )
9179, 90bitrd 261 . . . . . . 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 265 . . . . . 6  |-  ( ph  ->  ( ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  <->  ( z  e.  D  /\  a  =  <. I ,  J >. ) ) )
9392opabbidv 4459 . . . . 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 6313 . . . . . . . 8  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
953, 94eqtri 2493 . . . . . . 7  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
9695cnveqi 5014 . . . . . 6  |-  `' F  =  `' { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
97 nfv 1769 . . . . . . . 8  |-  F/ x  a  e.  ( A  X.  B )
98 nfcsb1v 3365 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
9998nfeq2 2627 . . . . . . . 8  |-  F/ x  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
10097, 99nfan 2031 . . . . . . 7  |-  F/ x
( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
101 nfv 1769 . . . . . . . 8  |-  F/ y  a  e.  ( A  X.  B )
102 nfcv 2612 . . . . . . . . . 10  |-  F/_ y
( 1st `  a
)
103 nfcsb1v 3365 . . . . . . . . . 10  |-  F/_ y [_ ( 2nd `  a
)  /  y ]_ C
104102, 103nfcsb 3367 . . . . . . . . 9  |-  F/_ y [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
105104nfeq2 2627 . . . . . . . 8  |-  F/ y  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  /  y ]_ C
106101, 105nfan 2031 . . . . . . 7  |-  F/ y ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
107 eleq1 2537 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
108 opelxp 4869 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
109107, 108syl6bb 269 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
110 csbopeq1a 6865 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C  =  C )
111110eqeq2d 2481 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( z  = 
[_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C 
<->  z  =  C ) )
112109, 111anbi12d 725 . . . . . . 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 4946 . . . . . . . . 9  |-  ( A  X.  B )  C_  ( _V  X.  _V )
114113sseli 3414 . . . . . . . 8  |-  ( a  e.  ( A  X.  B )  ->  a  e.  ( _V  X.  _V ) )
115114adantr 472 . . . . . . 7  |-  ( ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  ->  a  e.  ( _V  X.  _V ) )
116100, 106, 112, 115cnvoprab 28383 . . . . . 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 2493 . . . . 5  |-  `' F  =  { <. z ,  a
>.  |  ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) }
118 df-mpt 4456 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  { <. z ,  a >.  |  ( z  e.  D  /\  a  = 
<. I ,  J >. ) }
11993, 117, 1183eqtr4g 2530 . . . 4  |-  ( ph  ->  `' F  =  (
z  e.  D  |->  <.
I ,  J >. ) )
120119fneq1d 5676 . . 3  |-  ( ph  ->  ( `' F  Fn  D 
<->  ( z  e.  D  |-> 
<. I ,  J >. )  Fn  D ) )
12112, 120mpbird 240 . 2  |-  ( ph  ->  `' F  Fn  D
)
122 dff1o4 5836 . 2  |-  ( F : ( A  X.  B ) -1-1-onto-> D  <->  ( F  Fn  ( A  X.  B
)  /\  `' F  Fn  D ) )
1235, 121, 122sylanbrc 677 1  |-  ( ph  ->  F : ( A  X.  B ) -1-1-onto-> D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031   [.wsbc 3255   [_csb 3349   <.cop 3965   {copab 4453    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838    Fn wfn 5584   -1-1-onto->wf1o 5588   ` cfv 5589   {coprab 6309    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813
This theorem is referenced by:  oddpwdc  29260
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