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Theorem mapsnen 6954
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 5899 . 2  |-  ( A  ^m  { B }
)  e.  _V
2 mapsnen.1 . 2  |-  A  e. 
_V
3 fvex 5555 . . 3  |-  ( z `
 B )  e. 
_V
43a1i 10 . 2  |-  ( z  e.  ( A  ^m  { B } )  -> 
( z `  B
)  e.  _V )
5 snex 4232 . . 3  |-  { <. B ,  w >. }  e.  _V
65a1i 10 . 2  |-  ( w  e.  A  ->  { <. B ,  w >. }  e.  _V )
7 mapsnen.2 . . . . . . 7  |-  B  e. 
_V
82, 7mapsn 6825 . . . . . 6  |-  ( A  ^m  { B }
)  =  { z  |  E. y  e.  A  z  =  { <. B ,  y >. } }
98abeq2i 2403 . . . . 5  |-  ( z  e.  ( A  ^m  { B } )  <->  E. y  e.  A  z  =  { <. B ,  y
>. } )
109anbi1i 676 . . . 4  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
11 r19.41v 2706 . . . 4  |-  ( E. y  e.  A  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
12 df-rex 2562 . . . 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 264 . . 3  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  E. y
( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) ) )
14 fveq1 5540 . . . . . . . . . 10  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  ( { <. B ,  y >. } `  B ) )
15 vex 2804 . . . . . . . . . . 11  |-  y  e. 
_V
167, 15fvsn 5729 . . . . . . . . . 10  |-  ( {
<. B ,  y >. } `  B )  =  y
1714, 16syl6eq 2344 . . . . . . . . 9  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  y )
1817eqeq2d 2307 . . . . . . . 8  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  w  =  y ) )
19 equcom 1665 . . . . . . . 8  |-  ( w  =  y  <->  y  =  w )
2018, 19syl6bb 252 . . . . . . 7  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  y  =  w ) )
2120pm5.32i 618 . . . . . 6  |-  ( ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( z  =  { <. B ,  y
>. }  /\  y  =  w ) )
2221anbi2i 675 . . . . 5  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
23 anass 630 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
24 ancom 437 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
2522, 23, 243bitr2i 264 . . . 4  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
2625exbii 1572 . . 3  |-  ( E. y ( y  e.  A  /\  ( z  =  { <. B , 
y >. }  /\  w  =  ( z `  B ) ) )  <->  E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) ) )
27 vex 2804 . . . 4  |-  w  e. 
_V
28 eleq1 2356 . . . . 5  |-  ( y  =  w  ->  (
y  e.  A  <->  w  e.  A ) )
29 opeq2 3813 . . . . . . 7  |-  ( y  =  w  ->  <. B , 
y >.  =  <. B ,  w >. )
3029sneqd 3666 . . . . . 6  |-  ( y  =  w  ->  { <. B ,  y >. }  =  { <. B ,  w >. } )
3130eqeq2d 2307 . . . . 5  |-  ( y  =  w  ->  (
z  =  { <. B ,  y >. }  <->  z  =  { <. B ,  w >. } ) )
3228, 31anbi12d 691 . . . 4  |-  ( y  =  w  ->  (
( y  e.  A  /\  z  =  { <. B ,  y >. } )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) ) )
3327, 32ceqsexv 2836 . . 3  |-  ( E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
3413, 26, 333bitri 262 . 2  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
351, 2, 4, 6, 34en2i 6915 1  |-  ( A  ^m  { B }
)  ~~  A
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   {csn 3653   <.cop 3656   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    ^m cmap 6788    ~~ cen 6876
This theorem is referenced by:  map2xp  7047  mapdom3  7049  ackbij1lem5  7866  pwxpndom2  8303  hashmap  11403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-en 6880
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