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Theorem map2xp 7684
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 7140 . . . . 5  |-  2o  =  { (/) ,  1o }
2 df-pr 4030 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
31, 2eqtri 2496 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
43oveq2i 6293 . . 3  |-  ( A  ^m  2o )  =  ( A  ^m  ( { (/) }  u.  { 1o } ) )
5 snex 4688 . . . . 5  |-  { (/) }  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
7 snex 4688 . . . . 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 7142 . . . . . . . 8  |-  1o  =/=  (/)
1110neii 2666 . . . . . . 7  |-  -.  1o  =  (/)
12 elsni 4052 . . . . . . 7  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1311, 12mto 176 . . . . . 6  |-  -.  1o  e.  { (/) }
14 disjsn 4088 . . . . . 6  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  { (/) } )
1513, 14mpbir 209 . . . . 5  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
1615a1i 11 . . . 4  |-  ( A  e.  V  ->  ( { (/) }  i^i  { 1o } )  =  (/) )
17 mapunen 7683 . . . 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 1231 . . 3  |-  ( A  e.  V  ->  ( A  ^m  ( { (/) }  u.  { 1o }
) )  ~~  (
( A  ^m  { (/)
} )  X.  ( A  ^m  { 1o }
) ) )
194, 18syl5eqbr 4480 . 2  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
20 oveq1 6289 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { (/) } )  =  ( A  ^m  {
(/) } ) )
21 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
2220, 21breq12d 4460 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { (/)
} )  ~~  x  <->  ( A  ^m  { (/) } )  ~~  A ) )
23 vex 3116 . . . . 5  |-  x  e. 
_V
24 0ex 4577 . . . . 5  |-  (/)  e.  _V
2523, 24mapsnen 7590 . . . 4  |-  ( x  ^m  { (/) } ) 
~~  x
2622, 25vtoclg 3171 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { (/) } ) 
~~  A )
27 oveq1 6289 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { 1o } )  =  ( A  ^m  { 1o } ) )
2827, 21breq12d 4460 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { 1o } )  ~~  x  <->  ( A  ^m  { 1o } )  ~~  A
) )
29 df1o2 7139 . . . . . 6  |-  1o  =  { (/) }
3029, 5eqeltri 2551 . . . . 5  |-  1o  e.  _V
3123, 30mapsnen 7590 . . . 4  |-  ( x  ^m  { 1o }
)  ~~  x
3228, 31vtoclg 3171 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { 1o }
)  ~~  A )
33 xpen 7677 . . 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 7564 . 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 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   {cpr 4029   class class class wbr 4447    X. cxp 4997  (class class class)co 6282   1oc1o 7120   2oc2o 7121    ^m cmap 7417    ~~ cen 7510
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 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-1o 7127  df-2o 7128  df-er 7308  df-map 7419  df-en 7514  df-dom 7515
This theorem is referenced by:  pwxpndom2  9039
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