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Theorem map2xp 7706
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 7161 . . . . 5  |-  2o  =  { (/) ,  1o }
2 df-pr 4035 . . . . 5  |-  { (/) ,  1o }  =  ( { (/) }  u.  { 1o } )
31, 2eqtri 2486 . . . 4  |-  2o  =  ( { (/) }  u.  { 1o } )
43oveq2i 6307 . . 3  |-  ( A  ^m  2o )  =  ( A  ^m  ( { (/) }  u.  { 1o } ) )
5 snex 4697 . . . . 5  |-  { (/) }  e.  _V
65a1i 11 . . . 4  |-  ( A  e.  V  ->  { (/) }  e.  _V )
7 snex 4697 . . . . 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 7163 . . . . . . . 8  |-  1o  =/=  (/)
1110neii 2656 . . . . . . 7  |-  -.  1o  =  (/)
12 elsni 4057 . . . . . . 7  |-  ( 1o  e.  { (/) }  ->  1o  =  (/) )
1311, 12mto 176 . . . . . 6  |-  -.  1o  e.  { (/) }
14 disjsn 4092 . . . . . 6  |-  ( ( { (/) }  i^i  { 1o } )  =  (/)  <->  -.  1o  e.  { (/) } )
1513, 14mpbir 209 . . . . 5  |-  ( {
(/) }  i^i  { 1o } )  =  (/)
1615a1i 11 . . . 4  |-  ( A  e.  V  ->  ( { (/) }  i^i  { 1o } )  =  (/) )
17 mapunen 7705 . . . 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 4489 . 2  |-  ( A  e.  V  ->  ( A  ^m  2o )  ~~  ( ( A  ^m  {
(/) } )  X.  ( A  ^m  { 1o }
) ) )
20 oveq1 6303 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { (/) } )  =  ( A  ^m  {
(/) } ) )
21 id 22 . . . . 5  |-  ( x  =  A  ->  x  =  A )
2220, 21breq12d 4469 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { (/)
} )  ~~  x  <->  ( A  ^m  { (/) } )  ~~  A ) )
23 vex 3112 . . . . 5  |-  x  e. 
_V
24 0ex 4587 . . . . 5  |-  (/)  e.  _V
2523, 24mapsnen 7612 . . . 4  |-  ( x  ^m  { (/) } ) 
~~  x
2622, 25vtoclg 3167 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { (/) } ) 
~~  A )
27 oveq1 6303 . . . . 5  |-  ( x  =  A  ->  (
x  ^m  { 1o } )  =  ( A  ^m  { 1o } ) )
2827, 21breq12d 4469 . . . 4  |-  ( x  =  A  ->  (
( x  ^m  { 1o } )  ~~  x  <->  ( A  ^m  { 1o } )  ~~  A
) )
29 df1o2 7160 . . . . . 6  |-  1o  =  { (/) }
3029, 5eqeltri 2541 . . . . 5  |-  1o  e.  _V
3123, 30mapsnen 7612 . . . 4  |-  ( x  ^m  { 1o }
)  ~~  x
3228, 31vtoclg 3167 . . 3  |-  ( A  e.  V  ->  ( A  ^m  { 1o }
)  ~~  A )
33 xpen 7699 . . 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 7586 . 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 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    i^i cin 3470   (/)c0 3793   {csn 4032   {cpr 4034   class class class wbr 4456    X. cxp 5006  (class class class)co 6296   1oc1o 7141   2oc2o 7142    ^m cmap 7438    ~~ cen 7532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-1o 7148  df-2o 7149  df-er 7329  df-map 7440  df-en 7536  df-dom 7537
This theorem is referenced by:  pwxpndom2  9060
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