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Theorem xpsff1o 14502
Description: The function appearing in xpsval 14506 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair  2o  =  { (/)
,  1o }. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
xpsff1o.f  |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } ) )
Assertion
Ref Expression
xpsff1o  |-  F :
( A  X.  B
)
-1-1-onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
Distinct variable groups:    x, k,
y, A    B, k, x, y
Allowed substitution hints:    F( x, y, k)

Proof of Theorem xpsff1o
Dummy variables  a 
b  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsfrnel2 14499 . . . . . 6  |-  ( `' ( { x }  +c  { y } )  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
( x  e.  A  /\  y  e.  B
) )
21biimpri 206 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  B )  ->  `' ( { x }  +c  { y } )  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B ) )
32rgen2 2810 . . . 4  |-  A. x  e.  A  A. y  e.  B  `' ( { x }  +c  { y } )  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
4 xpsff1o.f . . . . 5  |-  F  =  ( x  e.  A ,  y  e.  B  |->  `' ( { x }  +c  { y } ) )
54fmpt2 6640 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  `' ( { x }  +c  { y } )  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
F : ( A  X.  B ) --> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B ) )
63, 5mpbi 208 . . 3  |-  F :
( A  X.  B
) --> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
7 1st2nd2 6612 . . . . . . . 8  |-  ( z  e.  ( A  X.  B )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
87fveq2d 5692 . . . . . . 7  |-  ( z  e.  ( A  X.  B )  ->  ( F `  z )  =  ( F `  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
)
9 df-ov 6093 . . . . . . . 8  |-  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
10 xp1st 6605 . . . . . . . . 9  |-  ( z  e.  ( A  X.  B )  ->  ( 1st `  z )  e.  A )
11 xp2nd 6606 . . . . . . . . 9  |-  ( z  e.  ( A  X.  B )  ->  ( 2nd `  z )  e.  B )
124xpsfval 14501 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  e.  A  /\  ( 2nd `  z )  e.  B )  -> 
( ( 1st `  z
) F ( 2nd `  z ) )  =  `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) )
1310, 11, 12syl2anc 656 . . . . . . . 8  |-  ( z  e.  ( A  X.  B )  ->  (
( 1st `  z
) F ( 2nd `  z ) )  =  `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) )
149, 13syl5eqr 2487 . . . . . . 7  |-  ( z  e.  ( A  X.  B )  ->  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )  =  `' ( { ( 1st `  z
) }  +c  {
( 2nd `  z
) } ) )
158, 14eqtrd 2473 . . . . . 6  |-  ( z  e.  ( A  X.  B )  ->  ( F `  z )  =  `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) )
16 1st2nd2 6612 . . . . . . . 8  |-  ( w  e.  ( A  X.  B )  ->  w  =  <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
1716fveq2d 5692 . . . . . . 7  |-  ( w  e.  ( A  X.  B )  ->  ( F `  w )  =  ( F `  <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
18 df-ov 6093 . . . . . . . 8  |-  ( ( 1st `  w ) F ( 2nd `  w
) )  =  ( F `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
19 xp1st 6605 . . . . . . . . 9  |-  ( w  e.  ( A  X.  B )  ->  ( 1st `  w )  e.  A )
20 xp2nd 6606 . . . . . . . . 9  |-  ( w  e.  ( A  X.  B )  ->  ( 2nd `  w )  e.  B )
214xpsfval 14501 . . . . . . . . 9  |-  ( ( ( 1st `  w
)  e.  A  /\  ( 2nd `  w )  e.  B )  -> 
( ( 1st `  w
) F ( 2nd `  w ) )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) )
2219, 20, 21syl2anc 656 . . . . . . . 8  |-  ( w  e.  ( A  X.  B )  ->  (
( 1st `  w
) F ( 2nd `  w ) )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) )
2318, 22syl5eqr 2487 . . . . . . 7  |-  ( w  e.  ( A  X.  B )  ->  ( F `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )  =  `' ( { ( 1st `  w
) }  +c  {
( 2nd `  w
) } ) )
2417, 23eqtrd 2473 . . . . . 6  |-  ( w  e.  ( A  X.  B )  ->  ( F `  w )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) )
2515, 24eqeqan12d 2456 . . . . 5  |-  ( ( z  e.  ( A  X.  B )  /\  w  e.  ( A  X.  B ) )  -> 
( ( F `  z )  =  ( F `  w )  <->  `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } )  =  `' ( { ( 1st `  w
) }  +c  {
( 2nd `  w
) } ) ) )
26 fveq1 5687 . . . . . . . 8  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) `  (/) )  =  ( `' ( { ( 1st `  w
) }  +c  {
( 2nd `  w
) } ) `  (/) ) )
27 fvex 5698 . . . . . . . . 9  |-  ( 1st `  z )  e.  _V
28 xpsc0 14494 . . . . . . . . 9  |-  ( ( 1st `  z )  e.  _V  ->  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) `  (/) )  =  ( 1st `  z
) )
2927, 28ax-mp 5 . . . . . . . 8  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } ) `  (/) )  =  ( 1st `  z )
30 fvex 5698 . . . . . . . . 9  |-  ( 1st `  w )  e.  _V
31 xpsc0 14494 . . . . . . . . 9  |-  ( ( 1st `  w )  e.  _V  ->  ( `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) `  (/) )  =  ( 1st `  w
) )
3230, 31ax-mp 5 . . . . . . . 8  |-  ( `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w
) } ) `  (/) )  =  ( 1st `  w )
3326, 29, 323eqtr3g 2496 . . . . . . 7  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  ( 1st `  z )  =  ( 1st `  w
) )
34 fveq1 5687 . . . . . . . 8  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) `  1o )  =  ( `' ( { ( 1st `  w
) }  +c  {
( 2nd `  w
) } ) `  1o ) )
35 fvex 5698 . . . . . . . . 9  |-  ( 2nd `  z )  e.  _V
36 xpsc1 14495 . . . . . . . . 9  |-  ( ( 2nd `  z )  e.  _V  ->  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) `  1o )  =  ( 2nd `  z ) )
3735, 36ax-mp 5 . . . . . . . 8  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } ) `  1o )  =  ( 2nd `  z )
38 fvex 5698 . . . . . . . . 9  |-  ( 2nd `  w )  e.  _V
39 xpsc1 14495 . . . . . . . . 9  |-  ( ( 2nd `  w )  e.  _V  ->  ( `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) `  1o )  =  ( 2nd `  w ) )
4038, 39ax-mp 5 . . . . . . . 8  |-  ( `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w
) } ) `  1o )  =  ( 2nd `  w )
4134, 37, 403eqtr3g 2496 . . . . . . 7  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  ( 2nd `  z )  =  ( 2nd `  w
) )
4233, 41opeq12d 4064 . . . . . 6  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
437, 16eqeqan12d 2456 . . . . . 6  |-  ( ( z  e.  ( A  X.  B )  /\  w  e.  ( A  X.  B ) )  -> 
( z  =  w  <->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. ) )
4442, 43syl5ibr 221 . . . . 5  |-  ( ( z  e.  ( A  X.  B )  /\  w  e.  ( A  X.  B ) )  -> 
( `' ( { ( 1st `  z
) }  +c  {
( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  z  =  w ) )
4525, 44sylbid 215 . . . 4  |-  ( ( z  e.  ( A  X.  B )  /\  w  e.  ( A  X.  B ) )  -> 
( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
4645rgen2 2810 . . 3  |-  A. z  e.  ( A  X.  B
) A. w  e.  ( A  X.  B
) ( ( F `
 z )  =  ( F `  w
)  ->  z  =  w )
47 dff13 5968 . . 3  |-  ( F : ( A  X.  B ) -1-1-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <->  ( F : ( A  X.  B ) --> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  /\  A. z  e.  ( A  X.  B ) A. w  e.  ( A  X.  B
) ( ( F `
 z )  =  ( F `  w
)  ->  z  =  w ) ) )
486, 46, 47mpbir2an 906 . 2  |-  F :
( A  X.  B
) -1-1-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
49 xpsfrnel 14497 . . . . . 6  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
( z  Fn  2o  /\  ( z `  (/) )  e.  A  /\  ( z `
 1o )  e.  B ) )
5049simp2bi 999 . . . . 5  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  ( z `  (/) )  e.  A )
5149simp3bi 1000 . . . . 5  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  ( z `  1o )  e.  B
)
524xpsfval 14501 . . . . . . 7  |-  ( ( ( z `  (/) )  e.  A  /\  ( z `
 1o )  e.  B )  ->  (
( z `  (/) ) F ( z `  1o ) )  =  `' ( { ( z `  (/) ) }  +c  {
( z `  1o ) } ) )
5350, 51, 52syl2anc 656 . . . . . 6  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  ( ( z `
 (/) ) F ( z `  1o ) )  =  `' ( { ( z `  (/) ) }  +c  {
( z `  1o ) } ) )
54 ixpfn 7265 . . . . . . 7  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  z  Fn  2o )
55 xpsfeq 14498 . . . . . . 7  |-  ( z  Fn  2o  ->  `' ( { ( z `  (/) ) }  +c  {
( z `  1o ) } )  =  z )
5654, 55syl 16 . . . . . 6  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  `' ( { ( z `  (/) ) }  +c  { ( z `
 1o ) } )  =  z )
5753, 56eqtr2d 2474 . . . . 5  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  z  =  ( ( z `  (/) ) F ( z `  1o ) ) )
58 rspceov 6127 . . . . 5  |-  ( ( ( z `  (/) )  e.  A  /\  ( z `
 1o )  e.  B  /\  z  =  ( ( z `  (/) ) F ( z `
 1o ) ) )  ->  E. a  e.  A  E. b  e.  B  z  =  ( a F b ) )
5950, 51, 57, 58syl3anc 1213 . . . 4  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  E. a  e.  A  E. b  e.  B  z  =  ( a F b ) )
6059rgen 2779 . . 3  |-  A. z  e.  X_  k  e.  2o  if ( k  =  (/) ,  A ,  B ) E. a  e.  A  E. b  e.  B  z  =  ( a F b )
61 foov 6236 . . 3  |-  ( F : ( A  X.  B ) -onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <->  ( F : ( A  X.  B ) --> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  /\  A. z  e.  X_  k  e.  2o  if ( k  =  (/) ,  A ,  B ) E. a  e.  A  E. b  e.  B  z  =  ( a F b ) ) )
626, 60, 61mpbir2an 906 . 2  |-  F :
( A  X.  B
) -onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
63 df-f1o 5422 . 2  |-  ( F : ( A  X.  B ) -1-1-onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
( F : ( A  X.  B )
-1-1->
X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  /\  F : ( A  X.  B )
-onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B ) ) )
6448, 62, 63mpbir2an 906 1  |-  F :
( A  X.  B
)
-1-1-onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   E.wrex 2714   _Vcvv 2970   (/)c0 3634   ifcif 3788   {csn 3874   <.cop 3880    X. cxp 4834   `'ccnv 4835    Fn wfn 5410   -->wf 5411   -1-1->wf1 5412   -onto->wfo 5413   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1stc1st 6574   2ndc2nd 6575   1oc1o 6909   2oc2o 6910   X_cixp 7259    +c ccda 8332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-cda 8333
This theorem is referenced by:  xpsfrn  14503  xpsff1o2  14505
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