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Theorem map1 6935
Description: Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)
Assertion
Ref Expression
map1  |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem map1
StepHypRef Expression
1 ovex 5845 . . 3  |-  ( 1o 
^m  A )  e. 
_V
21a1i 12 . 2  |-  ( A  e.  V  ->  ( 1o  ^m  A )  e. 
_V )
3 df1o2 6487 . . . 4  |-  1o  =  { (/) }
4 p0ex 4197 . . . 4  |-  { (/) }  e.  _V
53, 4eqeltri 2355 . . 3  |-  1o  e.  _V
65a1i 12 . 2  |-  ( A  e.  V  ->  1o  e.  _V )
7 0ex 4152 . . 3  |-  (/)  e.  _V
87a1ii 26 . 2  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  ->  (/) 
e.  _V ) )
9 xpexg 4800 . . . 4  |-  ( ( A  e.  V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
104, 9mpan2 654 . . 3  |-  ( A  e.  V  ->  ( A  X.  { (/) } )  e.  _V )
1110a1d 24 . 2  |-  ( A  e.  V  ->  (
y  e.  1o  ->  ( A  X.  { (/) } )  e.  _V )
)
12 el1o 6494 . . . . 5  |-  ( y  e.  1o  <->  y  =  (/) )
1312a1i 12 . . . 4  |-  ( A  e.  V  ->  (
y  e.  1o  <->  y  =  (/) ) )
143oveq1i 5830 . . . . . . 7  |-  ( 1o 
^m  A )  =  ( { (/) }  ^m  A )
1514eleq2i 2349 . . . . . 6  |-  ( x  e.  ( 1o  ^m  A )  <->  x  e.  ( { (/) }  ^m  A
) )
16 elmapg 6781 . . . . . . 7  |-  ( ( { (/) }  e.  _V  /\  A  e.  V )  ->  ( x  e.  ( { (/) }  ^m  A )  <->  x : A
--> { (/) } ) )
174, 16mpan 653 . . . . . 6  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }  ^m  A )  <-> 
x : A --> { (/) } ) )
1815, 17syl5bb 250 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  <->  x : A
--> { (/) } ) )
197fconst2 5692 . . . . 5  |-  ( x : A --> { (/) }  <-> 
x  =  ( A  X.  { (/) } ) )
2018, 19syl6rbb 255 . . . 4  |-  ( A  e.  V  ->  (
x  =  ( A  X.  { (/) } )  <-> 
x  e.  ( 1o 
^m  A ) ) )
2113, 20anbi12d 693 . . 3  |-  ( A  e.  V  ->  (
( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) )  <->  ( y  =  (/)  /\  x  e.  ( 1o  ^m  A ) ) ) )
22 ancom 439 . . 3  |-  ( ( y  =  (/)  /\  x  e.  ( 1o  ^m  A
) )  <->  ( x  e.  ( 1o  ^m  A
)  /\  y  =  (/) ) )
2321, 22syl6rbb 255 . 2  |-  ( A  e.  V  ->  (
( x  e.  ( 1o  ^m  A )  /\  y  =  (/) ) 
<->  ( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) ) ) )
242, 6, 8, 11, 23en2d 6893 1  |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2790   (/)c0 3457   {csn 3642   class class class wbr 4025    X. cxp 4687   -->wf 5218  (class class class)co 5820   1oc1o 6468    ^m cmap 6768    ~~ cen 6856
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1o 6475  df-map 6770  df-en 6860
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