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Theorem map2xp 7214
Description: A cardinal power with exponent 2 is equivalent to a cross 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 6674 . . . . 5  |-  2o  =  { (/) ,  1o }
2 df-pr 3765 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
31, 2eqtri 2408 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
43oveq2i 6032 . . 3  |-  ( A  ^m  2o )  =  ( A  ^m  ( { (/) }  u.  { 1o } ) )
5 snex 4347 . . . . 5  |-  { (/) }  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
7 snex 4347 . . . . 5  |-  { 1o }  e.  _V
87a1i 11 . . . 4  |-  ( A  e.  V  ->  { 1o }  e.  _V )
9 id 20 . . . 4  |-  ( A  e.  V  ->  A  e.  V )
10 1n0 6676 . . . . . . . 8  |-  1o  =/=  (/)
11 df-ne 2553 . . . . . . . 8  |-  ( 1o  =/=  (/)  <->  -.  1o  =  (/) )
1210, 11mpbi 200 . . . . . . 7  |-  -.  1o  =  (/)
13 elsni 3782 . . . . . . 7  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1412, 13mto 169 . . . . . 6  |-  -.  1o  e.  { (/) }
15 disjsn 3812 . . . . . 6  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  { (/) } )
1614, 15mpbir 201 . . . . 5  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
1716a1i 11 . . . 4  |-  ( A  e.  V  ->  ( { (/) }  i^i  { 1o } )  =  (/) )
18 mapunen 7213 . . . 4  |-  ( ( ( { (/) }  e.  _V  /\  { 1o }  e.  _V  /\  A  e.  V )  /\  ( { (/) }  i^i  { 1o } )  =  (/) )  ->  ( A  ^m  ( { (/) }  u.  { 1o } ) )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
196, 8, 9, 17, 18syl31anc 1187 . . 3  |-  ( A  e.  V  ->  ( A  ^m  ( { (/) }  u.  { 1o }
) )  ~~  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) ) )
204, 19syl5eqbr 4187 . 2  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
21 oveq1 6028 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { (/) } )  =  ( A  ^m  {
(/) } ) )
22 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
2321, 22breq12d 4167 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { (/)
} )  ~~  x  <->  ( A  ^m  { (/) } )  ~~  A ) )
24 vex 2903 . . . . 5  |-  x  e. 
_V
25 0ex 4281 . . . . 5  |-  (/)  e.  _V
2624, 25mapsnen 7121 . . . 4  |-  ( x  ^m  { (/) } ) 
~~  x
2723, 26vtoclg 2955 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { (/) } ) 
~~  A )
28 oveq1 6028 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { 1o } )  =  ( A  ^m  { 1o } ) )
2928, 22breq12d 4167 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { 1o } )  ~~  x  <->  ( A  ^m  { 1o } )  ~~  A
) )
30 df1o2 6673 . . . . . 6  |-  1o  =  { (/) }
3130, 5eqeltri 2458 . . . . 5  |-  1o  e.  _V
3224, 31mapsnen 7121 . . . 4  |-  ( x  ^m  { 1o }
)  ~~  x
3329, 32vtoclg 2955 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { 1o }
)  ~~  A )
34 xpen 7207 . . 3  |-  ( ( ( A  ^m  { (/)
} )  ~~  A  /\  ( A  ^m  { 1o } )  ~~  A
)  ->  ( ( A  ^m  { (/) } )  X.  ( A  ^m  { 1o } ) ) 
~~  ( A  X.  A ) )
3527, 33, 34syl2anc 643 . 2  |-  ( A  e.  V  ->  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) )  ~~  ( A  X.  A ) )
36 entr 7096 . 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 ) )
3720, 35, 36syl2anc 643 1  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( A  X.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900    u. cun 3262    i^i cin 3263   (/)c0 3572   {csn 3758   {cpr 3759   class class class wbr 4154    X. cxp 4817  (class class class)co 6021   1oc1o 6654   2oc2o 6655    ^m cmap 6955    ~~ cen 7043
This theorem is referenced by:  pwxpndom2  8474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-1o 6661  df-2o 6662  df-er 6842  df-map 6957  df-en 7047  df-dom 7048
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