Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  repcpwti Unicode version

Theorem repcpwti 25264
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 2804 . . . . 5  |-  f  e. 
_V
21elixp 6839 . . . 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 5355 . . . . . 6  |-  ( f  Fn  A  <->  f  Fn  { I ,  J }
)
54a1i 10 . . . . 5  |-  ( I  =/=  J  ->  (
f  Fn  A  <->  f  Fn  { I ,  J }
) )
63raleqi 2753 . . . . . . 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 2810 . . . . . . . 8  |-  I  e. 
_V
9 repcpwti.4 . . . . . . . . 9  |-  J  e.  D
109elexi 2810 . . . . . . . 8  |-  J  e. 
_V
11 fveq2 5541 . . . . . . . . 9  |-  ( x  =  I  ->  (
f `  x )  =  ( f `  I ) )
12 repcpwti.2 . . . . . . . . . 10  |-  B  =  if ( x  =  I ,  M ,  N )
13 iftrue 3584 . . . . . . . . . 10  |-  ( x  =  I  ->  if ( x  =  I ,  M ,  N )  =  M )
1412, 13syl5eq 2340 . . . . . . . . 9  |-  ( x  =  I  ->  B  =  M )
1511, 14eleq12d 2364 . . . . . . . 8  |-  ( x  =  I  ->  (
( f `  x
)  e.  B  <->  ( f `  I )  e.  M
) )
16 fveq2 5541 . . . . . . . . 9  |-  ( x  =  J  ->  (
f `  x )  =  ( f `  J ) )
17 eqeq1 2302 . . . . . . . . . . 11  |-  ( x  =  J  ->  (
x  =  I  <->  J  =  I ) )
1817ifbid 3596 . . . . . . . . . 10  |-  ( x  =  J  ->  if ( x  =  I ,  M ,  N )  =  if ( J  =  I ,  M ,  N ) )
1912, 18syl5eq 2340 . . . . . . . . 9  |-  ( x  =  J  ->  B  =  if ( J  =  I ,  M ,  N ) )
2016, 19eleq12d 2364 . . . . . . . 8  |-  ( x  =  J  ->  (
( f `  x
)  e.  B  <->  ( f `  J )  e.  if ( J  =  I ,  M ,  N ) ) )
218, 10, 15, 20ralpr 3699 . . . . . . 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 2540 . . . . . . . . . 10  |-  ( I  =/=  J  <->  J  =/=  I )
24 df-ne 2461 . . . . . . . . . 10  |-  ( J  =/=  I  <->  -.  J  =  I )
2523, 24bitri 240 . . . . . . . . 9  |-  ( I  =/=  J  <->  -.  J  =  I )
26 iffalse 3585 . . . . . . . . 9  |-  ( -.  J  =  I  ->  if ( J  =  I ,  M ,  N
)  =  N )
2725, 26sylbi 187 . . . . . . . 8  |-  ( I  =/=  J  ->  if ( J  =  I ,  M ,  N )  =  N )
2827eleq2d 2363 . . . . . . 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 25263 . . . . . . 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 3813 . . . . . . . . . 10  |-  ( a  =  ( f `  I )  ->  <. I ,  a >.  =  <. I ,  ( f `  I ) >. )
3635preq1d 3725 . . . . . . . . 9  |-  ( a  =  ( f `  I )  ->  { <. I ,  a >. ,  <. J ,  b >. }  =  { <. I ,  ( f `  I )
>. ,  <. J , 
b >. } )
3736eqeq2d 2307 . . . . . . . 8  |-  ( a  =  ( f `  I )  ->  (
f  =  { <. I ,  a >. ,  <. J ,  b >. }  <->  f  =  { <. I ,  ( f `  I )
>. ,  <. J , 
b >. } ) )
38 opeq2 3813 . . . . . . . . . 10  |-  ( b  =  ( f `  J )  ->  <. J , 
b >.  =  <. J , 
( f `  J
) >. )
3938preq2d 3726 . . . . . . . . 9  |-  ( b  =  ( f `  J )  ->  { <. I ,  ( f `  I ) >. ,  <. J ,  b >. }  =  { <. I ,  ( f `  I )
>. ,  <. J , 
( f `  J
) >. } )
4039eqeq2d 2307 . . . . . . . 8  |-  ( b  =  ( f `  J )  ->  (
f  =  { <. I ,  ( f `  I ) >. ,  <. J ,  b >. }  <->  f  =  { <. I ,  ( f `  I )
>. ,  <. J , 
( f `  J
) >. } ) )
4137, 40rspc2ev 2905 . . . . . . 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 2804 . . . . . . . . . 10  |-  a  e. 
_V
47 vex 2804 . . . . . . . . . 10  |-  b  e. 
_V
4846, 47pm3.2i 441 . . . . . . . . 9  |-  ( a  e.  _V  /\  b  e.  _V )
49 fnprg 5321 . . . . . . . . 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 25218 . . . . . . . . 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 2370 . . . . . . 7  |-  ( ( I  =/=  J  /\  ( a  e.  M  /\  b  e.  N
) )  ->  ( { <. I ,  a
>. ,  <. J , 
b >. } `  I
)  e.  M )
567, 9, 46, 47fvsn2b 25219 . . . . . . . . 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 2370 . . . . . . 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 5349 . . . . . . 7  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
f  Fn  { I ,  J }  <->  { <. I ,  a >. ,  <. J , 
b >. }  Fn  {
I ,  J }
) )
62 fveq1 5540 . . . . . . . . 9  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
f `  I )  =  ( { <. I ,  a >. ,  <. J ,  b >. } `  I ) )
6362eleq1d 2362 . . . . . . . 8  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
( f `  I
)  e.  M  <->  ( { <. I ,  a >. ,  <. J ,  b
>. } `  I )  e.  M ) )
64 fveq1 5540 . . . . . . . . 9  |-  ( f  =  { <. I ,  a >. ,  <. J , 
b >. }  ->  (
f `  J )  =  ( { <. I ,  a >. ,  <. J ,  b >. } `  J ) )
6564eleq1d 2362 . . . . . . . 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 2687 . . . 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 2411 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 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801   ifcif 3578   {cpr 3654   <.cop 3656    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  cbcpcp  25265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ixp 6834
  Copyright terms: Public domain W3C validator