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Theorem map2xp 7486
Description: A cardinal power with exponent 2 is equivalent to a Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
Assertion
Ref Expression
map2xp  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A
) )

Proof of Theorem map2xp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df2o3 6938 . . . . 5  |-  2o  =  { (/) ,  1o }
2 df-pr 3885 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
31, 2eqtri 2463 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
43oveq2i 6107 . . 3  |-  ( A  ^m  2o )  =  ( A  ^m  ( { (/) }  u.  { 1o } ) )
5 snex 4538 . . . . 5  |-  { (/) }  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
7 snex 4538 . . . . 5  |-  { 1o }  e.  _V
87a1i 11 . . . 4  |-  ( A  e.  V  ->  { 1o }  e.  _V )
9 id 22 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
10 1n0 6940 . . . . . . . 8  |-  1o  =/=  (/)
1110neii 2615 . . . . . . 7  |-  -.  1o  =  (/)
12 elsni 3907 . . . . . . 7  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1311, 12mto 176 . . . . . 6  |-  -.  1o  e.  { (/) }
14 disjsn 3941 . . . . . 6  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  { (/) } )
1513, 14mpbir 209 . . . . 5  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
1615a1i 11 . . . 4  |-  ( A  e.  V  ->  ( { (/) }  i^i  { 1o } )  =  (/) )
17 mapunen 7485 . . . 4  |-  ( ( ( { (/) }  e.  _V  /\  { 1o }  e.  _V  /\  A  e.  V )  /\  ( { (/) }  i^i  { 1o } )  =  (/) )  ->  ( A  ^m  ( { (/) }  u.  { 1o } ) )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
186, 8, 9, 16, 17syl31anc 1221 . . 3  |-  ( A  e.  V  ->  ( A  ^m  ( { (/) }  u.  { 1o }
) )  ~~  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) ) )
194, 18syl5eqbr 4330 . 2  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
20 oveq1 6103 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { (/) } )  =  ( A  ^m  {
(/) } ) )
21 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
2220, 21breq12d 4310 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { (/)
} )  ~~  x  <->  ( A  ^m  { (/) } )  ~~  A ) )
23 vex 2980 . . . . 5  |-  x  e. 
_V
24 0ex 4427 . . . . 5  |-  (/)  e.  _V
2523, 24mapsnen 7392 . . . 4  |-  ( x  ^m  { (/) } ) 
~~  x
2622, 25vtoclg 3035 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { (/) } ) 
~~  A )
27 oveq1 6103 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { 1o } )  =  ( A  ^m  { 1o } ) )
2827, 21breq12d 4310 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { 1o } )  ~~  x  <->  ( A  ^m  { 1o } )  ~~  A
) )
29 df1o2 6937 . . . . . 6  |-  1o  =  { (/) }
3029, 5eqeltri 2513 . . . . 5  |-  1o  e.  _V
3123, 30mapsnen 7392 . . . 4  |-  ( x  ^m  { 1o }
)  ~~  x
3228, 31vtoclg 3035 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { 1o }
)  ~~  A )
33 xpen 7479 . . 3  |-  ( ( ( A  ^m  { (/)
} )  ~~  A  /\  ( A  ^m  { 1o } )  ~~  A
)  ->  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) ) 
~~  ( A  X.  A ) )
3426, 32, 33syl2anc 661 . 2  |-  ( A  e.  V  ->  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) )  ~~  ( A  X.  A ) )
35 entr 7366 . 2  |-  ( ( ( A  ^m  2o )  ~~  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) )  /\  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) ) 
~~  ( A  X.  A ) )  -> 
( A  ^m  2o )  ~~  ( A  X.  A ) )
3619, 34, 35syl2anc 661 1  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977    u. cun 3331    i^i cin 3332   (/)c0 3642   {csn 3882   {cpr 3884   class class class wbr 4297    X. cxp 4843  (class class class)co 6096   1oc1o 6918   2oc2o 6919    ^m cmap 7219    ~~ cen 7312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-1o 6925  df-2o 6926  df-er 7106  df-map 7221  df-en 7316  df-dom 7317
This theorem is referenced by:  pwxpndom2  8837
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