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Theorem xpsff1o 14822
Description: The function appearing in xpsval 14826 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 14819 . . . . . 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 2889 . . . 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 6851 . . . 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 6821 . . . . . . . 8  |-  ( z  e.  ( A  X.  B )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
87fveq2d 5869 . . . . . . 7  |-  ( z  e.  ( A  X.  B )  ->  ( F `  z )  =  ( F `  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
)
9 df-ov 6286 . . . . . . . 8  |-  ( ( 1st `  z ) F ( 2nd `  z
) )  =  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
10 xp1st 6814 . . . . . . . . 9  |-  ( z  e.  ( A  X.  B )  ->  ( 1st `  z )  e.  A )
11 xp2nd 6815 . . . . . . . . 9  |-  ( z  e.  ( A  X.  B )  ->  ( 2nd `  z )  e.  B )
124xpsfval 14821 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  e.  A  /\  ( 2nd `  z )  e.  B )  -> 
( ( 1st `  z
) F ( 2nd `  z ) )  =  `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) )
1310, 11, 12syl2anc 661 . . . . . . . 8  |-  ( z  e.  ( A  X.  B )  ->  (
( 1st `  z
) F ( 2nd `  z ) )  =  `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) )
149, 13syl5eqr 2522 . . . . . . 7  |-  ( z  e.  ( A  X.  B )  ->  ( F `  <. ( 1st `  z ) ,  ( 2nd `  z )
>. )  =  `' ( { ( 1st `  z
) }  +c  {
( 2nd `  z
) } ) )
158, 14eqtrd 2508 . . . . . 6  |-  ( z  e.  ( A  X.  B )  ->  ( F `  z )  =  `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) )
16 1st2nd2 6821 . . . . . . . 8  |-  ( w  e.  ( A  X.  B )  ->  w  =  <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
1716fveq2d 5869 . . . . . . 7  |-  ( w  e.  ( A  X.  B )  ->  ( F `  w )  =  ( F `  <. ( 1st `  w
) ,  ( 2nd `  w ) >. )
)
18 df-ov 6286 . . . . . . . 8  |-  ( ( 1st `  w ) F ( 2nd `  w
) )  =  ( F `  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
19 xp1st 6814 . . . . . . . . 9  |-  ( w  e.  ( A  X.  B )  ->  ( 1st `  w )  e.  A )
20 xp2nd 6815 . . . . . . . . 9  |-  ( w  e.  ( A  X.  B )  ->  ( 2nd `  w )  e.  B )
214xpsfval 14821 . . . . . . . . 9  |-  ( ( ( 1st `  w
)  e.  A  /\  ( 2nd `  w )  e.  B )  -> 
( ( 1st `  w
) F ( 2nd `  w ) )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) )
2219, 20, 21syl2anc 661 . . . . . . . 8  |-  ( w  e.  ( A  X.  B )  ->  (
( 1st `  w
) F ( 2nd `  w ) )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) )
2318, 22syl5eqr 2522 . . . . . . 7  |-  ( w  e.  ( A  X.  B )  ->  ( F `  <. ( 1st `  w ) ,  ( 2nd `  w )
>. )  =  `' ( { ( 1st `  w
) }  +c  {
( 2nd `  w
) } ) )
2417, 23eqtrd 2508 . . . . . 6  |-  ( w  e.  ( A  X.  B )  ->  ( F `  w )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } ) )
2515, 24eqeqan12d 2490 . . . . 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 5864 . . . . . . . 8  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z ) } ) `  (/) )  =  ( `' ( { ( 1st `  w
) }  +c  {
( 2nd `  w
) } ) `  (/) ) )
27 fvex 5875 . . . . . . . . 9  |-  ( 1st `  z )  e.  _V
28 xpsc0 14814 . . . . . . . . 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 5875 . . . . . . . . 9  |-  ( 1st `  w )  e.  _V
31 xpsc0 14814 . . . . . . . . 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 2531 . . . . . . 7  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  ( 1st `  z )  =  ( 1st `  w
) )
34 fveq1 5864 . . . . . . . 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 5875 . . . . . . . . 9  |-  ( 2nd `  z )  e.  _V
36 xpsc1 14815 . . . . . . . . 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 5875 . . . . . . . . 9  |-  ( 2nd `  w )  e.  _V
39 xpsc1 14815 . . . . . . . . 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 2531 . . . . . . 7  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  ( 2nd `  z )  =  ( 2nd `  w
) )
4233, 41opeq12d 4221 . . . . . 6  |-  ( `' ( { ( 1st `  z ) }  +c  { ( 2nd `  z
) } )  =  `' ( { ( 1st `  w ) }  +c  { ( 2nd `  w ) } )  ->  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  =  <. ( 1st `  w ) ,  ( 2nd `  w
) >. )
437, 16eqeqan12d 2490 . . . . . 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 2889 . . 3  |-  A. z  e.  ( A  X.  B
) A. w  e.  ( A  X.  B
) ( ( F `
 z )  =  ( F `  w
)  ->  z  =  w )
47 dff13 6153 . . 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 918 . 2  |-  F :
( A  X.  B
) -1-1-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
49 xpsfrnel 14817 . . . . . 6  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
( z  Fn  2o  /\  ( z `  (/) )  e.  A  /\  ( z `
 1o )  e.  B ) )
5049simp2bi 1012 . . . . 5  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  ( z `  (/) )  e.  A )
5149simp3bi 1013 . . . . 5  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  ( z `  1o )  e.  B
)
524xpsfval 14821 . . . . . . 7  |-  ( ( ( z `  (/) )  e.  A  /\  ( z `
 1o )  e.  B )  ->  (
( z `  (/) ) F ( z `  1o ) )  =  `' ( { ( z `  (/) ) }  +c  {
( z `  1o ) } ) )
5350, 51, 52syl2anc 661 . . . . . 6  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  ( ( z `
 (/) ) F ( z `  1o ) )  =  `' ( { ( z `  (/) ) }  +c  {
( z `  1o ) } ) )
54 ixpfn 7475 . . . . . . 7  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  z  Fn  2o )
55 xpsfeq 14818 . . . . . . 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 2509 . . . . 5  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  z  =  ( ( z `  (/) ) F ( z `  1o ) ) )
58 rspceov 6320 . . . . 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 1228 . . . 4  |-  ( z  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  ->  E. a  e.  A  E. b  e.  B  z  =  ( a F b ) )
6059rgen 2824 . . 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 6432 . . 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 918 . 2  |-  F :
( A  X.  B
) -onto-> X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )
63 df-f1o 5594 . 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 918 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   (/)c0 3785   ifcif 3939   {csn 4027   <.cop 4033    X. cxp 4997   `'ccnv 4998    Fn wfn 5582   -->wf 5583   -1-1->wf1 5584   -onto->wfo 5585   -1-1-onto->wf1o 5586   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   1stc1st 6782   2ndc2nd 6783   1oc1o 7123   2oc2o 7124   X_cixp 7469    +c ccda 8546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-cda 8547
This theorem is referenced by:  xpsfrn  14823  xpsff1o2  14825
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