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Theorem map1 6939
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 )

Proof of Theorem map1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 5883 . . 3  |-  ( 1o 
^m  A )  e. 
_V
21a1i 10 . 2  |-  ( A  e.  V  ->  ( 1o  ^m  A )  e. 
_V )
3 df1o2 6491 . . . 4  |-  1o  =  { (/) }
4 p0ex 4197 . . . 4  |-  { (/) }  e.  _V
53, 4eqeltri 2353 . . 3  |-  1o  e.  _V
65a1i 10 . 2  |-  ( A  e.  V  ->  1o  e.  _V )
7 0ex 4150 . . 3  |-  (/)  e.  _V
87a1ii 24 . 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 652 . . 3  |-  ( A  e.  V  ->  ( A  X.  { (/) } )  e.  _V )
1110a1d 22 . 2  |-  ( A  e.  V  ->  (
y  e.  1o  ->  ( A  X.  { (/) } )  e.  _V )
)
12 el1o 6498 . . . . 5  |-  ( y  e.  1o  <->  y  =  (/) )
1312a1i 10 . . . 4  |-  ( A  e.  V  ->  (
y  e.  1o  <->  y  =  (/) ) )
143oveq1i 5868 . . . . . . 7  |-  ( 1o 
^m  A )  =  ( { (/) }  ^m  A )
1514eleq2i 2347 . . . . . 6  |-  ( x  e.  ( 1o  ^m  A )  <->  x  e.  ( { (/) }  ^m  A
) )
16 elmapg 6785 . . . . . . 7  |-  ( ( { (/) }  e.  _V  /\  A  e.  V )  ->  ( x  e.  ( { (/) }  ^m  A )  <->  x : A
--> { (/) } ) )
174, 16mpan 651 . . . . . 6  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }  ^m  A )  <-> 
x : A --> { (/) } ) )
1815, 17syl5bb 248 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  <->  x : A
--> { (/) } ) )
197fconst2 5730 . . . . 5  |-  ( x : A --> { (/) }  <-> 
x  =  ( A  X.  { (/) } ) )
2018, 19syl6rbb 253 . . . 4  |-  ( A  e.  V  ->  (
x  =  ( A  X.  { (/) } )  <-> 
x  e.  ( 1o 
^m  A ) ) )
2113, 20anbi12d 691 . . 3  |-  ( A  e.  V  ->  (
( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) )  <->  ( y  =  (/)  /\  x  e.  ( 1o  ^m  A ) ) ) )
22 ancom 437 . . 3  |-  ( ( y  =  (/)  /\  x  e.  ( 1o  ^m  A
) )  <->  ( x  e.  ( 1o  ^m  A
)  /\  y  =  (/) ) )
2321, 22syl6rbb 253 . 2  |-  ( A  e.  V  ->  (
( x  e.  ( 1o  ^m  A )  /\  y  =  (/) ) 
<->  ( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) ) ) )
242, 6, 8, 11, 23en2d 6897 1  |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640   class class class wbr 4023    X. cxp 4687   -->wf 5251  (class class class)co 5858   1oc1o 6472    ^m cmap 6772    ~~ cen 6860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-map 6774  df-en 6864
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