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Theorem f1od2 26024
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 2808 . . 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 6642 . . 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 4871 . . . . . 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 2799 . . . 4  |-  ( ph  ->  A. z  e.  D  <. I ,  J >.  e.  ( X  X.  Y
) )
10 eqid 2443 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  ( z  e.  D  |->  <.
I ,  J >. )
1110fnmpt 5537 . . . 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 6609 . . . . . . . 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 3245 . . . . . . . . . . . 12  |-  ( ph  ->  ( [. ( 2nd `  a )  /  y ]. ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  [. ( 2nd `  a )  /  y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) ) ) )
1817sbcbidv 3245 . . . . . . . . . . 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 3229 . . . . . . . . . . . . . 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 3229 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  e.  A  /\  y  e.  B )  <->  (
[. ( 2nd `  a
)  /  y ]. x  e.  A  /\  [. ( 2nd `  a
)  /  y ]. y  e.  B )
)
21 fvex 5701 . . . . . . . . . . . . . . . . . 18  |-  ( 2nd `  a )  e.  _V
22 sbcg 3260 . . . . . . . . . . . . . . . . . 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 3251 . . . . . . . . . . . . . . . . 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 3685 . . . . . . . . . . . . . . . 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 3246 . . . . . . . . . . . 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 3229 . . . . . . . . . . . 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 3229 . . . . . . . . . . . . . 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 3251 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. x  e.  A  <->  ( 1st `  a
)  e.  A )
35 fvex 5701 . . . . . . . . . . . . . . . 16  |-  ( 1st `  a )  e.  _V
36 sbcg 3260 . . . . . . . . . . . . . . . 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 3685 . . . . . . . . . . . . . 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 3229 . . . . . . . . . . . . . 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 3260 . . . . . . . . . . . . . . . 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 3229 . . . . . . . . . . . . . . . 16  |-  ( [. ( 2nd `  a )  /  y ]. (
x  =  I  /\  y  =  J )  <->  (
[. ( 2nd `  a
)  /  y ]. x  =  I  /\  [. ( 2nd `  a
)  /  y ]. y  =  J )
)
48 sbcg 3260 . . . . . . . . . . . . . . . . . 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 3682 . . . . . . . . . . . . . . . . . . 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 3700 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  a )  e.  _V  ->  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a ) )
5321, 52ax-mp 5 . . . . . . . . . . . . . . . . . . 19  |-  [_ ( 2nd `  a )  / 
y ]_ y  =  ( 2nd `  a )
5453eqeq1i 2450 . . . . . . . . . . . . . . . . . 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 3246 . . . . . . . . . . . 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 3229 . . . . . . . . . . . . 13  |-  ( [. ( 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 3260 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. z  e.  D  <->  z  e.  D ) )
6335, 62ax-mp 5 . . . . . . . . . . . . . 14  |-  ( [. ( 1st `  a )  /  x ]. z  e.  D  <->  z  e.  D
)
64 sbcan 3229 . . . . . . . . . . . . . . 15  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( [. ( 1st `  a )  /  x ]. x  =  I  /\  [. ( 1st `  a )  /  x ]. ( 2nd `  a
)  =  J ) )
65 sbceq1g 3682 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. x  =  I  <->  [_ ( 1st `  a )  /  x ]_ x  =  I
) )
6635, 65ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( [. ( 1st `  a )  /  x ]. x  =  I  <->  [_ ( 1st `  a
)  /  x ]_ x  =  I )
67 csbvarg 3700 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  a )  e.  _V  ->  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a ) )
6835, 67ax-mp 5 . . . . . . . . . . . . . . . . . 18  |-  [_ ( 1st `  a )  /  x ]_ x  =  ( 1st `  a )
6968eqeq1i 2450 . . . . . . . . . . . . . . . . 17  |-  ( [_ ( 1st `  a )  /  x ]_ x  =  I  <->  ( 1st `  a
)  =  I )
7066, 69bitri 249 . . . . . . . . . . . . . . . 16  |-  ( [. ( 1st `  a )  /  x ]. x  =  I  <->  ( 1st `  a
)  =  I )
71 sbcg 3260 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  a )  e.  _V  ->  ( [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  =  J  <->  ( 2nd `  a )  =  J ) )
7235, 71ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( [. ( 1st `  a )  /  x ]. ( 2nd `  a )  =  J  <->  ( 2nd `  a
)  =  J )
7370, 72anbi12i 697 . . . . . . . . . . . . . . 15  |-  ( (
[. ( 1st `  a
)  /  x ]. x  =  I  /\  [. ( 1st `  a
)  /  x ]. ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7464, 73bitri 249 . . . . . . . . . . . . . 14  |-  ( [. ( 1st `  a )  /  x ]. (
x  =  I  /\  ( 2nd `  a )  =  J )  <->  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) )
7563, 74anbi12i 697 . . . . . . . . . . . . 13  |-  ( (
[. ( 1st `  a
)  /  x ]. z  e.  D  /\  [. ( 1st `  a
)  /  x ]. ( x  =  I  /\  ( 2nd `  a
)  =  J ) )  <->  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )
7661, 75bitri 249 . . . . . . . . . . . 12  |-  ( [. ( 1st `  a )  /  x ]. (
z  e.  D  /\  ( x  =  I  /\  ( 2nd `  a
)  =  J ) )  <->  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )
7760, 76bitri 249 . . . . . . . . . . 11  |-  ( [. ( 1st `  a )  /  x ]. [. ( 2nd `  a )  / 
y ]. ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) )
7818, 43, 773bitr3g 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 ) ) ) )
7978anbi2d 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 ) ) ) ) )
8015, 79syl5bb 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 ) ) ) ) )
81 xpss 4946 . . . . . . . . . . . 12  |-  ( X  X.  Y )  C_  ( _V  X.  _V )
82 simprr 756 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  =  <. I ,  J >. )
838adantrr 716 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  <. I ,  J >.  e.  ( X  X.  Y ) )
8482, 83eqeltrd 2517 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( X  X.  Y
) )
8581, 84sseldi 3354 . . . . . . . . . . 11  |-  ( (
ph  /\  ( z  e.  D  /\  a  =  <. I ,  J >. ) )  ->  a  e.  ( _V  X.  _V ) )
8685ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  ->  a  e.  ( _V  X.  _V )
) )
8786pm4.71rd 635 . . . . . . . . 9  |-  ( ph  ->  ( ( z  e.  D  /\  a  = 
<. I ,  J >. )  <-> 
( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  a  =  <. I ,  J >. )
) ) )
88 eqop 6616 . . . . . . . . . . 11  |-  ( a  e.  ( _V  X.  _V )  ->  ( a  =  <. I ,  J >.  <-> 
( ( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) )
8988anbi2d 703 . . . . . . . . . 10  |-  ( a  e.  ( _V  X.  _V )  ->  ( ( z  e.  D  /\  a  =  <. I ,  J >. )  <->  ( z  e.  D  /\  (
( 1st `  a
)  =  I  /\  ( 2nd `  a )  =  J ) ) ) )
9089pm5.32i 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 ) ) ) )
9187, 90syl6rbb 262 . . . . . . . 8  |-  ( ph  ->  ( ( a  e.  ( _V  X.  _V )  /\  ( z  e.  D  /\  ( ( 1st `  a )  =  I  /\  ( 2nd `  a )  =  J ) ) )  <-> 
( z  e.  D  /\  a  =  <. I ,  J >. )
) )
9280, 91bitrd 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 >. ) ) )
9314, 92syl5bb 257 . . . . . 6  |-  ( ph  ->  ( ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  <->  ( z  e.  D  /\  a  =  <. I ,  J >. ) ) )
9493opabbidv 4355 . . . . 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 >. ) } )
95 df-mpt2 6096 . . . . . . . 8  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
963, 95eqtri 2463 . . . . . . 7  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
9796cnveqi 5014 . . . . . 6  |-  `' F  =  `' { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
98 nfv 1673 . . . . . . . 8  |-  F/ x  a  e.  ( A  X.  B )
99 nfcv 2579 . . . . . . . . 9  |-  F/_ x
z
100 nfcv 2579 . . . . . . . . . 10  |-  F/_ x
( 1st `  a
)
101100nfcsb1 3303 . . . . . . . . 9  |-  F/_ x [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
10299, 101nfeq 2586 . . . . . . . 8  |-  F/ x  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
10398, 102nfan 1861 . . . . . . 7  |-  F/ x
( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
104 nfv 1673 . . . . . . . 8  |-  F/ y  a  e.  ( A  X.  B )
105 nfcv 2579 . . . . . . . . 9  |-  F/_ y
z
106 nfcv 2579 . . . . . . . . . 10  |-  F/_ y
( 1st `  a
)
107 nfcv 2579 . . . . . . . . . . 11  |-  F/_ y
( 2nd `  a
)
108107nfcsb1 3303 . . . . . . . . . 10  |-  F/_ y [_ ( 2nd `  a
)  /  y ]_ C
109106, 108nfcsb 3306 . . . . . . . . 9  |-  F/_ y [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C
110105, 109nfeq 2586 . . . . . . . 8  |-  F/ y  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  /  y ]_ C
111104, 110nfan 1861 . . . . . . 7  |-  F/ y ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a )  / 
y ]_ C )
112 eleq1 2503 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
113 opelxp 4869 . . . . . . . . 9  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
114112, 113syl6bb 261 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( a  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) ) )
115 csbopeq1a 6627 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>.  ->  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C  =  C )
116115eqeq2d 2454 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( z  = 
[_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C 
<->  z  =  C ) )
117114, 116anbi12d 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
) ) )
118 xpss 4946 . . . . . . . . 9  |-  ( A  X.  B )  C_  ( _V  X.  _V )
119118sseli 3352 . . . . . . . 8  |-  ( a  e.  ( A  X.  B )  ->  a  e.  ( _V  X.  _V ) )
120119adantr 465 . . . . . . 7  |-  ( ( a  e.  ( A  X.  B )  /\  z  =  [_ ( 1st `  a )  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C )  ->  a  e.  ( _V  X.  _V ) )
121103, 111, 117, 120cnvoprab 26023 . . . . . 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 ) }
12297, 121eqtri 2463 . . . . 5  |-  `' F  =  { <. z ,  a
>.  |  ( a  e.  ( A  X.  B
)  /\  z  =  [_ ( 1st `  a
)  /  x ]_ [_ ( 2nd `  a
)  /  y ]_ C ) }
123 df-mpt 4352 . . . . 5  |-  ( z  e.  D  |->  <. I ,  J >. )  =  { <. z ,  a >.  |  ( z  e.  D  /\  a  = 
<. I ,  J >. ) }
12494, 122, 1233eqtr4g 2500 . . . 4  |-  ( ph  ->  `' F  =  (
z  e.  D  |->  <.
I ,  J >. ) )
125124fneq1d 5501 . . 3  |-  ( ph  ->  ( `' F  Fn  D 
<->  ( z  e.  D  |-> 
<. I ,  J >. )  Fn  D ) )
12612, 125mpbird 232 . 2  |-  ( ph  ->  `' F  Fn  D
)
127 dff1o4 5649 . 2  |-  ( F : ( A  X.  B ) -1-1-onto-> D  <->  ( F  Fn  ( A  X.  B
)  /\  `' F  Fn  D ) )
1285, 126, 127sylanbrc 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 1369    e. wcel 1756   A.wral 2715   _Vcvv 2972   [.wsbc 3186   [_csb 3288   <.cop 3883   {copab 4349    e. cmpt 4350    X. cxp 4838   `'ccnv 4839    Fn wfn 5413   -1-1-onto->wf1o 5417   ` cfv 5418   {coprab 6092    e. cmpt2 6093   1stc1st 6575   2ndc2nd 6576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578
This theorem is referenced by:  oddpwdc  26737
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