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Theorem cbcpcp 25162
Description: The canonical bijection between a cross product and a cartesian product (whose set of indices is composed of two different elements). Bourbaki E.II.33 . (Contributed by FL, 26-Jun-2011.)
Hypotheses
Ref Expression
cbcpcp.1  |-  A  =  { I ,  J }
cbcpcp.2  |-  B  =  if ( x  =  I ,  M ,  N )
cbcpcp.3  |-  F  =  ( a  e.  M ,  b  e.  N  |->  { <. I ,  a
>. ,  <. J , 
b >. } )
cbcpcp.4  |-  I  e.  C
cbcpcp.5  |-  J  e.  D
Assertion
Ref Expression
cbcpcp  |-  ( I  =/=  J  ->  F : ( M  X.  N ) -1-1-onto-> X_ x  e.  A  B )
Distinct variable groups:    A, a,
b, x    B, a,
b    x, F    I, a,
b, x    J, a,
b, x    M, a,
b, x    N, a,
b, x
Allowed substitution hints:    B( x)    C( x, a, b)    D( x, a, b)    F( a, b)

Proof of Theorem cbcpcp
Dummy variables  f 
c  d  e  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prex 4217 . . . 4  |-  { <. I ,  a >. ,  <. J ,  b >. }  e.  _V
21rgen2w 2611 . . 3  |-  A. a  e.  M  A. b  e.  N  { <. I ,  a >. ,  <. J , 
b >. }  e.  _V
3 cbcpcp.3 . . . 4  |-  F  =  ( a  e.  M ,  b  e.  N  |->  { <. I ,  a
>. ,  <. J , 
b >. } )
43fnmpt2 6192 . . 3  |-  ( A. a  e.  M  A. b  e.  N  { <. I ,  a >. ,  <. J ,  b
>. }  e.  _V  ->  F  Fn  ( M  X.  N ) )
52, 4mp1i 11 . 2  |-  ( I  =/=  J  ->  F  Fn  ( M  X.  N
) )
6 cbcpcp.1 . . . 4  |-  A  =  { I ,  J }
7 cbcpcp.2 . . . 4  |-  B  =  if ( x  =  I ,  M ,  N )
8 cbcpcp.4 . . . 4  |-  I  e.  C
9 cbcpcp.5 . . . 4  |-  J  e.  D
106, 7, 8, 9repcpwti 25161 . . 3  |-  ( I  =/=  J  ->  X_ x  e.  A  B  =  { f  |  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } } )
113rnmpt2 5954 . . 3  |-  ran  F  =  { f  |  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } }
1210, 11syl6reqr 2334 . 2  |-  ( I  =/=  J  ->  ran  F  =  X_ x  e.  A  B )
13 elxp 4706 . . . . . 6  |-  ( x  e.  ( M  X.  N )  <->  E. e E. f ( x  = 
<. e ,  f >.  /\  ( e  e.  M  /\  f  e.  N
) ) )
14 elxp 4706 . . . . . . . . . . 11  |-  ( y  e.  ( M  X.  N )  <->  E. c E. d ( y  = 
<. c ,  d >.  /\  ( c  e.  M  /\  d  e.  N
) ) )
15 prex 4217 . . . . . . . . . . . . . . 15  |-  { <. I ,  e >. ,  <. J ,  f >. }  e.  _V
16 df-ov 5861 . . . . . . . . . . . . . . . . . 18  |-  ( e F f )  =  ( F `  <. e ,  f >. )
17 opeq2 3797 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  =  e  ->  <. I ,  a >.  =  <. I ,  e >. )
1817preq1d 3712 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  e  ->  { <. I ,  a >. ,  <. J ,  b >. }  =  { <. I ,  e
>. ,  <. J , 
b >. } )
19 opeq2 3797 . . . . . . . . . . . . . . . . . . . 20  |-  ( b  =  f  ->  <. J , 
b >.  =  <. J , 
f >. )
2019preq2d 3713 . . . . . . . . . . . . . . . . . . 19  |-  ( b  =  f  ->  { <. I ,  e >. ,  <. J ,  b >. }  =  { <. I ,  e
>. ,  <. J , 
f >. } )
2118, 20, 3ovmpt2g 5982 . . . . . . . . . . . . . . . . . 18  |-  ( ( e  e.  M  /\  f  e.  N  /\  {
<. I ,  e >. ,  <. J ,  f
>. }  e.  _V )  ->  ( e F f )  =  { <. I ,  e >. ,  <. J ,  f >. } )
22 eqtr 2300 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( F `  <. e ,  f >. )  =  ( e F f )  /\  (
e F f )  =  { <. I ,  e >. ,  <. J , 
f >. } )  -> 
( F `  <. e ,  f >. )  =  { <. I ,  e
>. ,  <. J , 
f >. } )
23 prex 4217 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { <. I ,  c >. ,  <. J ,  d >. }  e.  _V
24 df-ov 5861 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( c F d )  =  ( F `  <. c ,  d >. )
25 opeq2 3797 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a  =  c  ->  <. I ,  a >.  =  <. I ,  c >. )
2625preq1d 3712 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( a  =  c  ->  { <. I ,  a >. ,  <. J ,  b >. }  =  { <. I ,  c
>. ,  <. J , 
b >. } )
27 opeq2 3797 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( b  =  d  ->  <. J , 
b >.  =  <. J , 
d >. )
2827preq2d 3713 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( b  =  d  ->  { <. I ,  c >. ,  <. J ,  b >. }  =  { <. I ,  c
>. ,  <. J , 
d >. } )
2926, 28, 3ovmpt2g 5982 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( c  e.  M  /\  d  e.  N  /\  {
<. I ,  c >. ,  <. J ,  d
>. }  e.  _V )  ->  ( c F d )  =  { <. I ,  c >. ,  <. J ,  d >. } )
30 eqtr 2300 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( F `  <. c ,  d >. )  =  ( c F d )  /\  (
c F d )  =  { <. I ,  c >. ,  <. J , 
d >. } )  -> 
( F `  <. c ,  d >. )  =  { <. I ,  c
>. ,  <. J , 
d >. } )
31 opex 4237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  <. I ,  e >.  e.  _V
32 opex 4237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  <. J , 
f >.  e.  _V
33 opex 4237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  <. I ,  c >.  e.  _V
34 opex 4237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  <. J , 
d >.  e.  _V
3531, 32, 33, 34preq12b 3788 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( {
<. I ,  e >. ,  <. J ,  f
>. }  =  { <. I ,  c >. ,  <. J ,  d >. }  <->  ( ( <. I ,  e >.  =  <. I ,  c
>.  /\  <. J ,  f
>.  =  <. J , 
d >. )  \/  ( <. I ,  e >.  =  <. J ,  d
>.  /\  <. J ,  f
>.  =  <. I ,  c >. ) ) )
368elexi 2797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  I  e. 
_V
37 vex 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  e  e. 
_V
3836, 37opth 4245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( <.
I ,  e >.  =  <. I ,  c
>. 
<->  ( I  =  I  /\  e  =  c ) )
399elexi 2797 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  J  e. 
_V
40 vex 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  f  e. 
_V
4139, 40opth 4245 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( <. J ,  f >.  = 
<. J ,  d >.  <->  ( J  =  J  /\  f  =  d )
)
42 opeq12 3798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( e  =  c  /\  f  =  d )  -> 
<. e ,  f >.  =  <. c ,  d
>. )
4342ad2ant2l 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( I  =  I  /\  e  =  c )  /\  ( J  =  J  /\  f  =  d ) )  ->  <. e ,  f
>.  =  <. c ,  d >. )
4438, 41, 43syl2anb 465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( (
<. I ,  e >.  =  <. I ,  c
>.  /\  <. J ,  f
>.  =  <. J , 
d >. )  ->  <. e ,  f >.  =  <. c ,  d >. )
4544a1d 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( (
<. I ,  e >.  =  <. I ,  c
>.  /\  <. J ,  f
>.  =  <. J , 
d >. )  ->  (
I  =/=  J  ->  <. e ,  f >.  =  <. c ,  d
>. ) )
4636, 37opth1 4244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( <.
I ,  e >.  =  <. J ,  d
>.  ->  I  =  J )
4746a1d 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( <.
I ,  e >.  =  <. J ,  d
>.  ->  ( -.  <. e ,  f >.  =  <. c ,  d >.  ->  I  =  J ) )
4847necon1ad 2513 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( <.
I ,  e >.  =  <. J ,  d
>.  ->  ( I  =/= 
J  ->  <. e ,  f >.  =  <. c ,  d >. )
)
4948adantr 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( (
<. I ,  e >.  =  <. J ,  d
>.  /\  <. J ,  f
>.  =  <. I ,  c >. )  ->  (
I  =/=  J  ->  <. e ,  f >.  =  <. c ,  d
>. ) )
5045, 49jaoi 368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( <. I ,  e
>.  =  <. I ,  c >.  /\  <. J , 
f >.  =  <. J , 
d >. )  \/  ( <. I ,  e >.  =  <. J ,  d
>.  /\  <. J ,  f
>.  =  <. I ,  c >. ) )  -> 
( I  =/=  J  -> 
<. e ,  f >.  =  <. c ,  d
>. ) )
5135, 50sylbi 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( {
<. I ,  e >. ,  <. J ,  f
>. }  =  { <. I ,  c >. ,  <. J ,  d >. }  ->  ( I  =/=  J  ->  <. e ,  f >.  =  <. c ,  d
>. ) )
5251com12 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( I  =/=  J  ->  ( { <. I ,  e
>. ,  <. J , 
f >. }  =  { <. I ,  c >. ,  <. J ,  d
>. }  ->  <. e ,  f >.  =  <. c ,  d >. )
)
53 eqeq2 2292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( F `  <. c ,  d >. )  =  { <. I ,  c
>. ,  <. J , 
d >. }  ->  ( { <. I ,  e
>. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  <->  {
<. I ,  e >. ,  <. J ,  f
>. }  =  { <. I ,  c >. ,  <. J ,  d >. } ) )
5453imbi1d 308 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F `  <. c ,  d >. )  =  { <. I ,  c
>. ,  <. J , 
d >. }  ->  (
( { <. I ,  e >. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. )  <->  ( { <. I ,  e >. ,  <. J ,  f >. }  =  { <. I ,  c
>. ,  <. J , 
d >. }  ->  <. e ,  f >.  =  <. c ,  d >. )
) )
5552, 54syl5ibr 212 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F `  <. c ,  d >. )  =  { <. I ,  c
>. ,  <. J , 
d >. }  ->  (
I  =/=  J  -> 
( { <. I ,  e >. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) )
5630, 55syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( F `  <. c ,  d >. )  =  ( c F d )  /\  (
c F d )  =  { <. I ,  c >. ,  <. J , 
d >. } )  -> 
( I  =/=  J  ->  ( { <. I ,  e >. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) )
5756ex 423 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( F `  <. c ,  d >. )  =  ( c F d )  ->  (
( c F d )  =  { <. I ,  c >. ,  <. J ,  d >. }  ->  ( I  =/=  J  -> 
( { <. I ,  e >. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) ) )
5857eqcoms 2286 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( c F d )  =  ( F `  <. c ,  d >.
)  ->  ( (
c F d )  =  { <. I ,  c >. ,  <. J , 
d >. }  ->  (
I  =/=  J  -> 
( { <. I ,  e >. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) ) )
5924, 29, 58mpsyl 59 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( c  e.  M  /\  d  e.  N  /\  {
<. I ,  c >. ,  <. J ,  d
>. }  e.  _V )  ->  ( I  =/=  J  ->  ( { <. I ,  e >. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) )
6023, 59mp3an3 1266 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( c  e.  M  /\  d  e.  N )  ->  ( I  =/=  J  ->  ( { <. I ,  e >. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) )
61 eqeq1 2289 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F `  <. e ,  f >. )  =  { <. I ,  e
>. ,  <. J , 
f >. }  ->  (
( F `  <. e ,  f >. )  =  ( F `  <. c ,  d >.
)  <->  { <. I ,  e
>. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )
) )
6261imbi1d 308 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F `  <. e ,  f >. )  =  { <. I ,  e
>. ,  <. J , 
f >. }  ->  (
( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )  <->  ( { <. I ,  e
>. ,  <. J , 
f >. }  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) )
6362imbi2d 307 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F `  <. e ,  f >. )  =  { <. I ,  e
>. ,  <. J , 
f >. }  ->  (
( I  =/=  J  ->  ( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
)  <->  ( I  =/= 
J  ->  ( { <. I ,  e >. ,  <. J ,  f
>. }  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) ) )
6460, 63syl5ibr 212 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F `  <. e ,  f >. )  =  { <. I ,  e
>. ,  <. J , 
f >. }  ->  (
( c  e.  M  /\  d  e.  N
)  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) ) )
6522, 64syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( F `  <. e ,  f >. )  =  ( e F f )  /\  (
e F f )  =  { <. I ,  e >. ,  <. J , 
f >. } )  -> 
( ( c  e.  M  /\  d  e.  N )  ->  (
I  =/=  J  -> 
( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) ) )
6665ex 423 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  <. e ,  f >. )  =  ( e F f )  ->  (
( e F f )  =  { <. I ,  e >. ,  <. J ,  f >. }  ->  ( ( c  e.  M  /\  d  e.  N
)  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) ) ) )
6766eqcoms 2286 . . . . . . . . . . . . . . . . . 18  |-  ( ( e F f )  =  ( F `  <. e ,  f >.
)  ->  ( (
e F f )  =  { <. I ,  e >. ,  <. J , 
f >. }  ->  (
( c  e.  M  /\  d  e.  N
)  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) ) ) )
6816, 21, 67mpsyl 59 . . . . . . . . . . . . . . . . 17  |-  ( ( e  e.  M  /\  f  e.  N  /\  {
<. I ,  e >. ,  <. J ,  f
>. }  e.  _V )  ->  ( ( c  e.  M  /\  d  e.  N )  ->  (
I  =/=  J  -> 
( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) ) )
69683expia 1153 . . . . . . . . . . . . . . . 16  |-  ( ( e  e.  M  /\  f  e.  N )  ->  ( { <. I ,  e >. ,  <. J , 
f >. }  e.  _V  ->  ( ( c  e.  M  /\  d  e.  N )  ->  (
I  =/=  J  -> 
( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) ) ) )
7069com3l 75 . . . . . . . . . . . . . . 15  |-  ( {
<. I ,  e >. ,  <. J ,  f
>. }  e.  _V  ->  ( ( c  e.  M  /\  d  e.  N
)  ->  ( (
e  e.  M  /\  f  e.  N )  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) ) ) )
7115, 70ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( c  e.  M  /\  d  e.  N )  ->  ( ( e  e.  M  /\  f  e.  N )  ->  (
I  =/=  J  -> 
( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) ) )
72 fveq2 5525 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  <. c ,  d
>.  ->  ( F `  y )  =  ( F `  <. c ,  d >. )
)
7372eqeq2d 2294 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. c ,  d
>.  ->  ( ( F `
 <. e ,  f
>. )  =  ( F `  y )  <->  ( F `  <. e ,  f >. )  =  ( F `  <. c ,  d >.
) ) )
74 eqeq2 2292 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. c ,  d
>.  ->  ( <. e ,  f >.  =  y  <->  <. e ,  f >.  =  <. c ,  d
>. ) )
7573, 74imbi12d 311 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. c ,  d
>.  ->  ( ( ( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f >.  =  y )  <->  ( ( F `
 <. e ,  f
>. )  =  ( F `  <. c ,  d >. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) )
7675imbi2d 307 . . . . . . . . . . . . . . 15  |-  ( y  =  <. c ,  d
>.  ->  ( ( I  =/=  J  ->  (
( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f >.  =  y ) )  <->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  <. c ,  d >. )  -> 
<. e ,  f >.  =  <. c ,  d
>. ) ) ) )
7776imbi2d 307 . . . . . . . . . . . . . 14  |-  ( y  =  <. c ,  d
>.  ->  ( ( ( e  e.  M  /\  f  e.  N )  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >.
)  =  ( F `
 y )  ->  <. e ,  f >.  =  y ) ) )  <->  ( ( e  e.  M  /\  f  e.  N )  ->  (
I  =/=  J  -> 
( ( F `  <. e ,  f >.
)  =  ( F `
 <. c ,  d
>. )  ->  <. e ,  f >.  =  <. c ,  d >. )
) ) ) )
7871, 77syl5ibr 212 . . . . . . . . . . . . 13  |-  ( y  =  <. c ,  d
>.  ->  ( ( c  e.  M  /\  d  e.  N )  ->  (
( e  e.  M  /\  f  e.  N
)  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f
>.  =  y )
) ) ) )
7978imp 418 . . . . . . . . . . . 12  |-  ( ( y  =  <. c ,  d >.  /\  (
c  e.  M  /\  d  e.  N )
)  ->  ( (
e  e.  M  /\  f  e.  N )  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >.
)  =  ( F `
 y )  ->  <. e ,  f >.  =  y ) ) ) )
8079exlimivv 1667 . . . . . . . . . . 11  |-  ( E. c E. d ( y  =  <. c ,  d >.  /\  (
c  e.  M  /\  d  e.  N )
)  ->  ( (
e  e.  M  /\  f  e.  N )  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >.
)  =  ( F `
 y )  ->  <. e ,  f >.  =  y ) ) ) )
8114, 80sylbi 187 . . . . . . . . . 10  |-  ( y  e.  ( M  X.  N )  ->  (
( e  e.  M  /\  f  e.  N
)  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f
>.  =  y )
) ) )
8281com12 27 . . . . . . . . 9  |-  ( ( e  e.  M  /\  f  e.  N )  ->  ( y  e.  ( M  X.  N )  ->  ( I  =/= 
J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f
>.  =  y )
) ) )
83 fveq2 5525 . . . . . . . . . . . . 13  |-  ( x  =  <. e ,  f
>.  ->  ( F `  x )  =  ( F `  <. e ,  f >. )
)
8483eqeq1d 2291 . . . . . . . . . . . 12  |-  ( x  =  <. e ,  f
>.  ->  ( ( F `
 x )  =  ( F `  y
)  <->  ( F `  <. e ,  f >.
)  =  ( F `
 y ) ) )
85 eqeq1 2289 . . . . . . . . . . . 12  |-  ( x  =  <. e ,  f
>.  ->  ( x  =  y  <->  <. e ,  f
>.  =  y )
)
8684, 85imbi12d 311 . . . . . . . . . . 11  |-  ( x  =  <. e ,  f
>.  ->  ( ( ( F `  x )  =  ( F `  y )  ->  x  =  y )  <->  ( ( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f
>.  =  y )
) )
8786imbi2d 307 . . . . . . . . . 10  |-  ( x  =  <. e ,  f
>.  ->  ( ( I  =/=  J  ->  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
)  <->  ( I  =/= 
J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f
>.  =  y )
) ) )
8887imbi2d 307 . . . . . . . . 9  |-  ( x  =  <. e ,  f
>.  ->  ( ( y  e.  ( M  X.  N )  ->  (
I  =/=  J  -> 
( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )  <->  ( y  e.  ( M  X.  N
)  ->  ( I  =/=  J  ->  ( ( F `  <. e ,  f >. )  =  ( F `  y )  ->  <. e ,  f
>.  =  y )
) ) ) )
8982, 88syl5ibr 212 . . . . . . . 8  |-  ( x  =  <. e ,  f
>.  ->  ( ( e  e.  M  /\  f  e.  N )  ->  (
y  e.  ( M  X.  N )  -> 
( I  =/=  J  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) ) ) )
9089imp 418 . . . . . . 7  |-  ( ( x  =  <. e ,  f >.  /\  (
e  e.  M  /\  f  e.  N )
)  ->  ( y  e.  ( M  X.  N
)  ->  ( I  =/=  J  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) ) )
9190exlimivv 1667 . . . . . 6  |-  ( E. e E. f ( x  =  <. e ,  f >.  /\  (
e  e.  M  /\  f  e.  N )
)  ->  ( y  e.  ( M  X.  N
)  ->  ( I  =/=  J  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) ) )
9213, 91sylbi 187 . . . . 5  |-  ( x  e.  ( M  X.  N )  ->  (
y  e.  ( M  X.  N )  -> 
( I  =/=  J  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) ) )
9392imp 418 . . . 4  |-  ( ( x  e.  ( M  X.  N )  /\  y  e.  ( M  X.  N ) )  -> 
( I  =/=  J  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
9493com12 27 . . 3  |-  ( I  =/=  J  ->  (
( x  e.  ( M  X.  N )  /\  y  e.  ( M  X.  N ) )  ->  ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
9594ralrimivv 2634 . 2  |-  ( I  =/=  J  ->  A. x  e.  ( M  X.  N
) A. y  e.  ( M  X.  N
) ( ( F `
 x )  =  ( F `  y
)  ->  x  =  y ) )
96 dff1o6 5791 . 2  |-  ( F : ( M  X.  N ) -1-1-onto-> X_ x  e.  A  B 
<->  ( F  Fn  ( M  X.  N )  /\  ran  F  =  X_ x  e.  A  B  /\  A. x  e.  ( M  X.  N ) A. y  e.  ( M  X.  N ) ( ( F `  x )  =  ( F `  y )  ->  x  =  y ) ) )
975, 12, 95, 96syl3anbrc 1136 1  |-  ( I  =/=  J  ->  F : ( M  X.  N ) -1-1-onto-> X_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   ifcif 3565   {cpr 3641   <.cop 3643    X. cxp 4687   ran crn 4690    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   X_cixp 6817
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ixp 6818
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