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

Theorem mapsnen 7385
Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
mapsnen.1  |-  A  e. 
_V
mapsnen.2  |-  B  e. 
_V
Assertion
Ref Expression
mapsnen  |-  ( A  ^m  { B }
)  ~~  A

Proof of Theorem mapsnen
Dummy variables  y  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6114 . 2  |-  ( A  ^m  { B }
)  e.  _V
2 mapsnen.1 . 2  |-  A  e. 
_V
3 fvex 5699 . . 3  |-  ( z `
 B )  e. 
_V
43a1i 11 . 2  |-  ( z  e.  ( A  ^m  { B } )  -> 
( z `  B
)  e.  _V )
5 snex 4531 . . 3  |-  { <. B ,  w >. }  e.  _V
65a1i 11 . 2  |-  ( w  e.  A  ->  { <. B ,  w >. }  e.  _V )
7 mapsnen.2 . . . . . . 7  |-  B  e. 
_V
82, 7mapsn 7252 . . . . . 6  |-  ( A  ^m  { B }
)  =  { z  |  E. y  e.  A  z  =  { <. B ,  y >. } }
98abeq2i 2548 . . . . 5  |-  ( z  e.  ( A  ^m  { B } )  <->  E. y  e.  A  z  =  { <. B ,  y
>. } )
109anbi1i 695 . . . 4  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
11 r19.41v 2871 . . . 4  |-  ( E. y  e.  A  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
12 df-rex 2719 . . . 4  |-  ( E. y  e.  A  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  E. y
( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) ) )
1310, 11, 123bitr2i 273 . . 3  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  E. y
( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) ) )
14 fveq1 5688 . . . . . . . . . 10  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  ( { <. B ,  y >. } `  B ) )
15 vex 2973 . . . . . . . . . . 11  |-  y  e. 
_V
167, 15fvsn 5909 . . . . . . . . . 10  |-  ( {
<. B ,  y >. } `  B )  =  y
1714, 16syl6eq 2489 . . . . . . . . 9  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  y )
1817eqeq2d 2452 . . . . . . . 8  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  w  =  y ) )
19 equcom 1732 . . . . . . . 8  |-  ( w  =  y  <->  y  =  w )
2018, 19syl6bb 261 . . . . . . 7  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  y  =  w ) )
2120pm5.32i 637 . . . . . 6  |-  ( ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( z  =  { <. B ,  y
>. }  /\  y  =  w ) )
2221anbi2i 694 . . . . 5  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
23 anass 649 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
24 ancom 450 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
2522, 23, 243bitr2i 273 . . . 4  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
2625exbii 1634 . . 3  |-  ( E. y ( y  e.  A  /\  ( z  =  { <. B , 
y >. }  /\  w  =  ( z `  B ) ) )  <->  E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) ) )
27 vex 2973 . . . 4  |-  w  e. 
_V
28 eleq1 2501 . . . . 5  |-  ( y  =  w  ->  (
y  e.  A  <->  w  e.  A ) )
29 opeq2 4058 . . . . . . 7  |-  ( y  =  w  ->  <. B , 
y >.  =  <. B ,  w >. )
3029sneqd 3887 . . . . . 6  |-  ( y  =  w  ->  { <. B ,  y >. }  =  { <. B ,  w >. } )
3130eqeq2d 2452 . . . . 5  |-  ( y  =  w  ->  (
z  =  { <. B ,  y >. }  <->  z  =  { <. B ,  w >. } ) )
3228, 31anbi12d 710 . . . 4  |-  ( y  =  w  ->  (
( y  e.  A  /\  z  =  { <. B ,  y >. } )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) ) )
3327, 32ceqsexv 3007 . . 3  |-  ( E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
3413, 26, 333bitri 271 . 2  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
351, 2, 4, 6, 34en2i 7345 1  |-  ( A  ^m  { B }
)  ~~  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2714   _Vcvv 2970   {csn 3875   <.cop 3881   class class class wbr 4290   ` cfv 5416  (class class class)co 6089    ^m cmap 7212    ~~ cen 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-en 7309
This theorem is referenced by:  map2xp  7479  mapdom3  7481  ackbij1lem5  8391  pwxpndom2  8830  hashmap  12195
  Copyright terms: Public domain W3C validator