MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  map1 Unicode version

Theorem map1 7176
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 6097 . . 3  |-  ( 1o 
^m  A )  e. 
_V
21a1i 11 . 2  |-  ( A  e.  V  ->  ( 1o  ^m  A )  e. 
_V )
3 df1o2 6727 . . . 4  |-  1o  =  { (/) }
4 p0ex 4378 . . . 4  |-  { (/) }  e.  _V
53, 4eqeltri 2505 . . 3  |-  1o  e.  _V
65a1i 11 . 2  |-  ( A  e.  V  ->  1o  e.  _V )
7 0ex 4331 . . 3  |-  (/)  e.  _V
87a1ii 25 . 2  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  ->  (/) 
e.  _V ) )
9 xpexg 4980 . . . 4  |-  ( ( A  e.  V  /\  {
(/) }  e.  _V )  ->  ( A  X.  { (/) } )  e. 
_V )
104, 9mpan2 653 . . 3  |-  ( A  e.  V  ->  ( A  X.  { (/) } )  e.  _V )
1110a1d 23 . 2  |-  ( A  e.  V  ->  (
y  e.  1o  ->  ( A  X.  { (/) } )  e.  _V )
)
12 el1o 6734 . . . . 5  |-  ( y  e.  1o  <->  y  =  (/) )
1312a1i 11 . . . 4  |-  ( A  e.  V  ->  (
y  e.  1o  <->  y  =  (/) ) )
143oveq1i 6082 . . . . . . 7  |-  ( 1o 
^m  A )  =  ( { (/) }  ^m  A )
1514eleq2i 2499 . . . . . 6  |-  ( x  e.  ( 1o  ^m  A )  <->  x  e.  ( { (/) }  ^m  A
) )
16 elmapg 7022 . . . . . . 7  |-  ( ( { (/) }  e.  _V  /\  A  e.  V )  ->  ( x  e.  ( { (/) }  ^m  A )  <->  x : A
--> { (/) } ) )
174, 16mpan 652 . . . . . 6  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }  ^m  A )  <-> 
x : A --> { (/) } ) )
1815, 17syl5bb 249 . . . . 5  |-  ( A  e.  V  ->  (
x  e.  ( 1o 
^m  A )  <->  x : A
--> { (/) } ) )
197fconst2 5939 . . . . 5  |-  ( x : A --> { (/) }  <-> 
x  =  ( A  X.  { (/) } ) )
2018, 19syl6rbb 254 . . . 4  |-  ( A  e.  V  ->  (
x  =  ( A  X.  { (/) } )  <-> 
x  e.  ( 1o 
^m  A ) ) )
2113, 20anbi12d 692 . . 3  |-  ( A  e.  V  ->  (
( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) )  <->  ( y  =  (/)  /\  x  e.  ( 1o  ^m  A ) ) ) )
22 ancom 438 . . 3  |-  ( ( y  =  (/)  /\  x  e.  ( 1o  ^m  A
) )  <->  ( x  e.  ( 1o  ^m  A
)  /\  y  =  (/) ) )
2321, 22syl6rbb 254 . 2  |-  ( A  e.  V  ->  (
( x  e.  ( 1o  ^m  A )  /\  y  =  (/) ) 
<->  ( y  e.  1o  /\  x  =  ( A  X.  { (/) } ) ) ) )
242, 6, 8, 11, 23en2d 7134 1  |-  ( A  e.  V  ->  ( 1o  ^m  A )  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   {csn 3806   class class class wbr 4204    X. cxp 4867   -->wf 5441  (class class class)co 6072   1oc1o 6708    ^m cmap 7009    ~~ cen 7097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1o 6715  df-map 7011  df-en 7101
  Copyright terms: Public domain W3C validator