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Theorem map1 6674
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 6517 . . 3  |-  ^m  Fn  ( _V  X.  _V )
2 1oex 6289 . . 3  |-  1o  e.  _V
3 elex 2671 . . 3  |-  ( A  e.  V  ->  A  e.  _V )
4 fnovex 5772 . . 3  |-  ( (  ^m  Fn  ( _V 
X.  _V )  /\  1o  e.  _V  /\  A  e. 
_V )  ->  ( 1o  ^m  A )  e. 
_V )
51, 2, 3, 4mp3an12i 1304 . 2  |-  ( A  e.  V  ->  ( 1o  ^m  A )  e. 
_V )
62a1i 9 . 2  |-  ( A  e.  V  ->  1o  e.  _V )
7 0ex 4025 . . 3  |-  (/)  e.  _V
872a1i 27 . 2  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  ->  (/) 
e.  _V ) )
9 p0ex 4082 . . . 4  |-  { (/) }  e.  _V
10 xpexg 4623 . . . 4  |-  ( ( A  e.  V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
119, 10mpan2 421 . . 3  |-  ( A  e.  V  ->  ( A  X.  { (/) } )  e.  _V )
1211a1d 22 . 2  |-  ( A  e.  V  ->  (
y  e.  1o  ->  ( A  X.  { (/) } )  e.  _V )
)
13 el1o 6302 . . . . 5  |-  ( y  e.  1o  <->  y  =  (/) )
1413a1i 9 . . . 4  |-  ( A  e.  V  ->  (
y  e.  1o  <->  y  =  (/) ) )
15 df1o2 6294 . . . . . . . 8  |-  1o  =  { (/) }
1615oveq1i 5752 . . . . . . 7  |-  ( 1o 
^m  A )  =  ( { (/) }  ^m  A )
1716eleq2i 2184 . . . . . 6  |-  ( x  e.  ( 1o  ^m  A )  <->  x  e.  ( { (/) }  ^m  A
) )
18 elmapg 6523 . . . . . . 7  |-  ( ( { (/) }  e.  _V  /\  A  e.  V )  ->  ( x  e.  ( { (/) }  ^m  A )  <->  x : A
--> { (/) } ) )
199, 18mpan 420 . . . . . 6  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }  ^m  A )  <-> 
x : A --> { (/) } ) )
2017, 19syl5bb 191 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  <->  x : A
--> { (/) } ) )
217fconst2 5605 . . . . 5  |-  ( x : A --> { (/) }  <-> 
x  =  ( A  X.  { (/) } ) )
2220, 21syl6rbb 196 . . . 4  |-  ( A  e.  V  ->  (
x  =  ( A  X.  { (/) } )  <-> 
x  e.  ( 1o 
^m  A ) ) )
2314, 22anbi12d 464 . . 3  |-  ( A  e.  V  ->  (
( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) )  <->  ( y  =  (/)  /\  x  e.  ( 1o  ^m  A ) ) ) )
24 ancom 264 . . 3  |-  ( ( y  =  (/)  /\  x  e.  ( 1o  ^m  A
) )  <->  ( x  e.  ( 1o  ^m  A
)  /\  y  =  (/) ) )
2523, 24syl6rbb 196 . 2  |-  ( A  e.  V  ->  (
( x  e.  ( 1o  ^m  A )  /\  y  =  (/) ) 
<->  ( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) ) ) )
265, 6, 8, 12, 25en2d 6630 1  |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   (/)c0 3333   {csn 3497   class class class wbr 3899    X. cxp 4507    Fn wfn 5088   -->wf 5089  (class class class)co 5742   1oc1o 6274    ^m cmap 6510    ~~ cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-1o 6281  df-map 6512  df-en 6603
This theorem is referenced by: (None)
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