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Theorem map1 6519
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 fnmap 6402 . . 3  |-  ^m  Fn  ( _V  X.  _V )
2 1oex 6181 . . 3  |-  1o  e.  _V
3 elex 2630 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
4 fnovex 5674 . . 3  |-  ( (  ^m  Fn  ( _V 
X.  _V )  /\  1o  e.  _V  /\  A  e. 
_V )  ->  ( 1o  ^m  A )  e. 
_V )
51, 2, 3, 4mp3an12i 1277 . 2  |-  ( A  e.  V  ->  ( 1o  ^m  A )  e. 
_V )
62a1i 9 . 2  |-  ( A  e.  V  ->  1o  e.  _V )
7 0ex 3964 . . 3  |-  (/)  e.  _V
872a1i 27 . 2  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  ->  (/) 
e.  _V ) )
9 p0ex 4021 . . . 4  |-  { (/) }  e.  _V
10 xpexg 4548 . . . 4  |-  ( ( A  e.  V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
119, 10mpan2 416 . . 3  |-  ( A  e.  V  ->  ( A  X.  { (/) } )  e.  _V )
1211a1d 22 . 2  |-  ( A  e.  V  ->  (
y  e.  1o  ->  ( A  X.  { (/) } )  e.  _V )
)
13 el1o 6193 . . . . 5  |-  ( y  e.  1o  <->  y  =  (/) )
1413a1i 9 . . . 4  |-  ( A  e.  V  ->  (
y  e.  1o  <->  y  =  (/) ) )
15 df1o2 6186 . . . . . . . 8  |-  1o  =  { (/) }
1615oveq1i 5654 . . . . . . 7  |-  ( 1o 
^m  A )  =  ( { (/) }  ^m  A )
1716eleq2i 2154 . . . . . 6  |-  ( x  e.  ( 1o  ^m  A )  <->  x  e.  ( { (/) }  ^m  A
) )
18 elmapg 6408 . . . . . . 7  |-  ( ( { (/) }  e.  _V  /\  A  e.  V )  ->  ( x  e.  ( { (/) }  ^m  A )  <->  x : A
--> { (/) } ) )
199, 18mpan 415 . . . . . 6  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }  ^m  A )  <-> 
x : A --> { (/) } ) )
2017, 19syl5bb 190 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  <->  x : A
--> { (/) } ) )
217fconst2 5506 . . . . 5  |-  ( x : A --> { (/) }  <-> 
x  =  ( A  X.  { (/) } ) )
2220, 21syl6rbb 195 . . . 4  |-  ( A  e.  V  ->  (
x  =  ( A  X.  { (/) } )  <-> 
x  e.  ( 1o 
^m  A ) ) )
2314, 22anbi12d 457 . . 3  |-  ( A  e.  V  ->  (
( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) )  <->  ( y  =  (/)  /\  x  e.  ( 1o  ^m  A ) ) ) )
24 ancom 262 . . 3  |-  ( ( y  =  (/)  /\  x  e.  ( 1o  ^m  A
) )  <->  ( x  e.  ( 1o  ^m  A
)  /\  y  =  (/) ) )
2523, 24syl6rbb 195 . 2  |-  ( A  e.  V  ->  (
( x  e.  ( 1o  ^m  A )  /\  y  =  (/) ) 
<->  ( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) ) ) )
265, 6, 8, 12, 25en2d 6475 1  |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   _Vcvv 2619   (/)c0 3286   {csn 3444   class class class wbr 3843    X. cxp 4434    Fn wfn 5005   -->wf 5006  (class class class)co 5644   1oc1o 6166    ^m cmap 6395    ~~ cen 6445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-suc 4196  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-1o 6173  df-map 6397  df-en 6448
This theorem is referenced by: (None)
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