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Theorem repcpwti 25161
Description: A representation of a cartesian product with two indices. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
repcpwti.1  |-  A  =  { I ,  J }
repcpwti.2  |-  B  =  if ( x  =  I ,  M ,  N )
repcpwti.3  |-  I  e.  C
repcpwti.4  |-  J  e.  D
Assertion
Ref Expression
repcpwti  |-  ( I  =/=  J  ->  X_ x  e.  A  B  =  { f  |  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } } )
Distinct variable groups:    x, M    x, A, f, a, b   
x, I, f, a, b    x, J, f, a, b    x, N    B, f, a, b    M, a, b    N, a, b
Allowed substitution hints:    B( x)    C( x, f, a, b)    D( x, f, a, b)    M( f)    N( f)

Proof of Theorem repcpwti
StepHypRef Expression
1 vex 2791 . . . . 5  |-  f  e. 
_V
21elixp 6823 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
3 repcpwti.1 . . . . . . 7  |-  A  =  { I ,  J }
43fneq2i 5339 . . . . . 6  |-  ( f  Fn  A  <->  f  Fn  { I ,  J }
)
54a1i 10 . . . . 5  |-  ( I  =/=  J  ->  (
f  Fn  A  <->  f  Fn  { I ,  J }
) )
63raleqi 2740 . . . . . . 7  |-  ( A. x  e.  A  (
f `  x )  e.  B  <->  A. x  e.  {
I ,  J } 
( f `  x
)  e.  B )
7 repcpwti.3 . . . . . . . . 9  |-  I  e.  C
87elexi 2797 . . . . . . . 8  |-  I  e. 
_V
9 repcpwti.4 . . . . . . . . 9  |-  J  e.  D
109elexi 2797 . . . . . . . 8  |-  J  e. 
_V
11 fveq2 5525 . . . . . . . . 9  |-  ( x  =  I  ->  (
f `  x )  =  ( f `  I ) )
12 repcpwti.2 . . . . . . . . . 10  |-  B  =  if ( x  =  I ,  M ,  N )
13 iftrue 3571 . . . . . . . . . 10  |-  ( x  =  I  ->  if ( x  =  I ,  M ,  N )  =  M )
1412, 13syl5eq 2327 . . . . . . . . 9  |-  ( x  =  I  ->  B  =  M )
1511, 14eleq12d 2351 . . . . . . . 8  |-  ( x  =  I  ->  (
( f `  x
)  e.  B  <->  ( f `  I )  e.  M
) )
16 fveq2 5525 . . . . . . . . 9  |-  ( x  =  J  ->  (
f `  x )  =  ( f `  J ) )
17 eqeq1 2289 . . . . . . . . . . 11  |-  ( x  =  J  ->  (
x  =  I  <->  J  =  I ) )
1817ifbid 3583 . . . . . . . . . 10  |-  ( x  =  J  ->  if ( x  =  I ,  M ,  N )  =  if ( J  =  I ,  M ,  N ) )
1912, 18syl5eq 2327 . . . . . . . . 9  |-  ( x  =  J  ->  B  =  if ( J  =  I ,  M ,  N ) )
2016, 19eleq12d 2351 . . . . . . . 8  |-  ( x  =  J  ->  (
( f `  x
)  e.  B  <->  ( f `  J )  e.  if ( J  =  I ,  M ,  N ) ) )
218, 10, 15, 20ralpr 3686 . . . . . . 7  |-  ( A. x  e.  { I ,  J }  ( f `
 x )  e.  B  <->  ( ( f `
 I )  e.  M  /\  ( f `
 J )  e.  if ( J  =  I ,  M ,  N ) ) )
226, 21bitri 240 . . . . . 6  |-  ( A. x  e.  A  (
f `  x )  e.  B  <->  ( ( f `
 I )  e.  M  /\  ( f `
 J )  e.  if ( J  =  I ,  M ,  N ) ) )
23 necom 2527 . . . . . . . . . 10  |-  ( I  =/=  J  <->  J  =/=  I )
24 df-ne 2448 . . . . . . . . . 10  |-  ( J  =/=  I  <->  -.  J  =  I )
2523, 24bitri 240 . . . . . . . . 9  |-  ( I  =/=  J  <->  -.  J  =  I )
26 iffalse 3572 . . . . . . . . 9  |-  ( -.  J  =  I  ->  if ( J  =  I ,  M ,  N
)  =  N )
2725, 26sylbi 187 . . . . . . . 8  |-  ( I  =/=  J  ->  if ( J  =  I ,  M ,  N )  =  N )
2827eleq2d 2350 . . . . . . 7  |-  ( I  =/=  J  ->  (
( f `  J
)  e.  if ( J  =  I ,  M ,  N )  <-> 
( f `  J
)  e.  N ) )
2928anbi2d 684 . . . . . 6  |-  ( I  =/=  J  ->  (
( ( f `  I )  e.  M  /\  ( f `  J
)  e.  if ( J  =  I ,  M ,  N ) )  <->  ( ( f `
 I )  e.  M  /\  ( f `
 J )  e.  N ) ) )
3022, 29syl5bb 248 . . . . 5  |-  ( I  =/=  J  ->  ( A. x  e.  A  ( f `  x
)  e.  B  <->  ( (
f `  I )  e.  M  /\  (
f `  J )  e.  N ) ) )
315, 30anbi12d 691 . . . 4  |-  ( I  =/=  J  ->  (
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B )  <-> 
( f  Fn  {
I ,  J }  /\  ( ( f `  I )  e.  M  /\  ( f `  J
)  e.  N ) ) ) )
322, 31syl5bb 248 . . 3  |-  ( I  =/=  J  ->  (
f  e.  X_ x  e.  A  B  <->  ( f  Fn  { I ,  J }  /\  ( ( f `
 I )  e.  M  /\  ( f `
 J )  e.  N ) ) ) )
337, 9repfuntw 25160 . . . . . . 7  |-  ( I  =/=  J  ->  (
f  Fn  { I ,  J }  <->  f  =  { <. I ,  ( f `  I )
>. ,  <. J , 
( f `  J
) >. } ) )
3433biimpa 470 . . . . . 6  |-  ( ( I  =/=  J  /\  f  Fn  { I ,  J } )  -> 
f  =  { <. I ,  ( f `  I ) >. ,  <. J ,  ( f `  J ) >. } )
35 opeq2 3797 . . . . . . . . . 10  |-  ( a  =  ( f `  I )  ->  <. I ,  a >.  =  <. I ,  ( f `  I ) >. )
3635preq1d 3712 . . . . . . . . 9  |-  ( a  =  ( f `  I )  ->  { <. I ,  a >. ,  <. J ,  b >. }  =  { <. I ,  ( f `  I )
>. ,  <. J , 
b >. } )
3736eqeq2d 2294 . . . . . . . 8  |-  ( a  =  ( f `  I )  ->  (
f  =  { <. I ,  a >. ,  <. J ,  b >. }  <->  f  =  { <. I ,  ( f `  I )
>. ,  <. J , 
b >. } ) )
38 opeq2 3797 . . . . . . . . . 10  |-  ( b  =  ( f `  J )  ->  <. J , 
b >.  =  <. J , 
( f `  J
) >. )
3938preq2d 3713 . . . . . . . . 9  |-  ( b  =  ( f `  J )  ->  { <. I ,  ( f `  I ) >. ,  <. J ,  b >. }  =  { <. I ,  ( f `  I )
>. ,  <. J , 
( f `  J
) >. } )
4039eqeq2d 2294 . . . . . . . 8  |-  ( b  =  ( f `  J )  ->  (
f  =  { <. I ,  ( f `  I ) >. ,  <. J ,  b >. }  <->  f  =  { <. I ,  ( f `  I )
>. ,  <. J , 
( f `  J
) >. } ) )
4137, 40rspc2ev 2892 . . . . . . 7  |-  ( ( ( f `  I
)  e.  M  /\  ( f `  J
)  e.  N  /\  f  =  { <. I ,  ( f `  I
) >. ,  <. J , 
( f `  J
) >. } )  ->  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a >. ,  <. J , 
b >. } )
42413expia 1153 . . . . . 6  |-  ( ( ( f `  I
)  e.  M  /\  ( f `  J
)  e.  N )  ->  ( f  =  { <. I ,  ( f `  I )
>. ,  <. J , 
( f `  J
) >. }  ->  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } ) )
4334, 42syl5com 26 . . . . 5  |-  ( ( I  =/=  J  /\  f  Fn  { I ,  J } )  -> 
( ( ( f `
 I )  e.  M  /\  ( f `
 J )  e.  N )  ->  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } ) )
4443expimpd 586 . . . 4  |-  ( I  =/=  J  ->  (
( f  Fn  {
I ,  J }  /\  ( ( f `  I )  e.  M  /\  ( f `  J
)  e.  N ) )  ->  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } ) )
457, 9pm3.2i 441 . . . . . . . . 9  |-  ( I  e.  C  /\  J  e.  D )
46 vex 2791 . . . . . . . . . 10  |-  a  e. 
_V
47 vex 2791 . . . . . . . . . 10  |-  b  e. 
_V
4846, 47pm3.2i 441 . . . . . . . . 9  |-  ( a  e.  _V  /\  b  e.  _V )
49 fnprg 5305 . . . . . . . . 9  |-  ( ( ( I  e.  C  /\  J  e.  D
)  /\  ( a  e.  _V  /\  b  e. 
_V )  /\  I  =/=  J )  ->  { <. I ,  a >. ,  <. J ,  b >. }  Fn  { I ,  J }
)
5045, 48, 49mp3an12 1267 . . . . . . . 8  |-  ( I  =/=  J  ->  { <. I ,  a >. ,  <. J ,  b >. }  Fn  { I ,  J }
)
5150adantr 451 . . . . . . 7  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  { <. I ,  a >. ,  <. J ,  b >. }  Fn  { I ,  J }
)
527, 9, 46, 47fvsn2a 25115 . . . . . . . . 9  |-  ( I  =/=  J  ->  ( { <. I ,  a
>. ,  <. J , 
b >. } `  I
)  =  a )
5352adantr 451 . . . . . . . 8  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  ( { <. I ,  a
>. ,  <. J , 
b >. } `  I
)  =  a )
54 simprl 732 . . . . . . . 8  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  a  e.  M )
5553, 54eqeltrd 2357 . . . . . . 7  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  ( { <. I ,  a
>. ,  <. J , 
b >. } `  I
)  e.  M )
567, 9, 46, 47fvsn2b 25116 . . . . . . . . 9  |-  ( I  =/=  J  ->  ( { <. I ,  a
>. ,  <. J , 
b >. } `  J
)  =  b )
5756adantr 451 . . . . . . . 8  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  ( { <. I ,  a
>. ,  <. J , 
b >. } `  J
)  =  b )
58 simprr 733 . . . . . . . 8  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  b  e.  N )
5957, 58eqeltrd 2357 . . . . . . 7  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  ( { <. I ,  a
>. ,  <. J , 
b >. } `  J
)  e.  N )
6051, 55, 59jca32 521 . . . . . 6  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  ( { <. I ,  a
>. ,  <. J , 
b >. }  Fn  {
I ,  J }  /\  ( ( { <. I ,  a >. ,  <. J ,  b >. } `  I )  e.  M  /\  ( { <. I ,  a >. ,  <. J , 
b >. } `  J
)  e.  N ) ) )
61 fneq1 5333 . . . . . . 7  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
f  Fn  { I ,  J }  <->  { <. I ,  a >. ,  <. J , 
b >. }  Fn  {
I ,  J }
) )
62 fveq1 5524 . . . . . . . . 9  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
f `  I )  =  ( { <. I ,  a >. ,  <. J ,  b >. } `  I ) )
6362eleq1d 2349 . . . . . . . 8  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
( f `  I
)  e.  M  <->  ( { <. I ,  a >. ,  <. J ,  b
>. } `  I )  e.  M ) )
64 fveq1 5524 . . . . . . . . 9  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
f `  J )  =  ( { <. I ,  a >. ,  <. J ,  b >. } `  J ) )
6564eleq1d 2349 . . . . . . . 8  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
( f `  J
)  e.  N  <->  ( { <. I ,  a >. ,  <. J ,  b
>. } `  J )  e.  N ) )
6663, 65anbi12d 691 . . . . . . 7  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
( ( f `  I )  e.  M  /\  ( f `  J
)  e.  N )  <-> 
( ( { <. I ,  a >. ,  <. J ,  b >. } `  I )  e.  M  /\  ( { <. I ,  a >. ,  <. J , 
b >. } `  J
)  e.  N ) ) )
6761, 66anbi12d 691 . . . . . 6  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
( f  Fn  {
I ,  J }  /\  ( ( f `  I )  e.  M  /\  ( f `  J
)  e.  N ) )  <->  ( { <. I ,  a >. ,  <. J ,  b >. }  Fn  { I ,  J }  /\  ( ( { <. I ,  a >. ,  <. J ,  b >. } `  I )  e.  M  /\  ( { <. I ,  a >. ,  <. J , 
b >. } `  J
)  e.  N ) ) ) )
6860, 67syl5ibrcom 213 . . . . 5  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  (
f  =  { <. I ,  a >. ,  <. J ,  b >. }  ->  ( f  Fn  { I ,  J }  /\  (
( f `  I
)  e.  M  /\  ( f `  J
)  e.  N ) ) ) )
6968rexlimdvva 2674 . . . 4  |-  ( I  =/=  J  ->  ( E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
f  Fn  { I ,  J }  /\  (
( f `  I
)  e.  M  /\  ( f `  J
)  e.  N ) ) ) )
7044, 69impbid 183 . . 3  |-  ( I  =/=  J  ->  (
( f  Fn  {
I ,  J }  /\  ( ( f `  I )  e.  M  /\  ( f `  J
)  e.  N ) )  <->  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a >. ,  <. J , 
b >. } ) )
7132, 70bitrd 244 . 2  |-  ( I  =/=  J  ->  (
f  e.  X_ x  e.  A  B  <->  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } ) )
7271abbi2dv 2398 1  |-  ( I  =/=  J  ->  X_ x  e.  A  B  =  { f  |  E. a  e.  M  E. b  e.  N  f  =  { <. I ,  a
>. ,  <. J , 
b >. } } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   ifcif 3565   {cpr 3641   <.cop 3643    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  cbcpcp  25162
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
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-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-fv 5263  df-ixp 6818
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