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Theorem mapsnen 7175
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 6097 . 2  |-  ( A  ^m  { B }
)  e.  _V
2 mapsnen.1 . 2  |-  A  e. 
_V
3 fvex 5733 . . 3  |-  ( z `
 B )  e. 
_V
43a1i 11 . 2  |-  ( z  e.  ( A  ^m  { B } )  -> 
( z `  B
)  e.  _V )
5 snex 4397 . . 3  |-  { <. B ,  w >. }  e.  _V
65a1i 11 . 2  |-  ( w  e.  A  ->  { <. B ,  w >. }  e.  _V )
7 mapsnen.2 . . . . . . 7  |-  B  e. 
_V
82, 7mapsn 7046 . . . . . 6  |-  ( A  ^m  { B }
)  =  { z  |  E. y  e.  A  z  =  { <. B ,  y >. } }
98abeq2i 2542 . . . . 5  |-  ( z  e.  ( A  ^m  { B } )  <->  E. y  e.  A  z  =  { <. B ,  y
>. } )
109anbi1i 677 . . . 4  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
11 r19.41v 2853 . . . 4  |-  ( E. y  e.  A  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
12 df-rex 2703 . . . 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 265 . . 3  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  E. y
( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) ) )
14 fveq1 5718 . . . . . . . . . 10  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  ( { <. B ,  y >. } `  B ) )
15 vex 2951 . . . . . . . . . . 11  |-  y  e. 
_V
167, 15fvsn 5917 . . . . . . . . . 10  |-  ( {
<. B ,  y >. } `  B )  =  y
1714, 16syl6eq 2483 . . . . . . . . 9  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  y )
1817eqeq2d 2446 . . . . . . . 8  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  w  =  y ) )
19 equcom 1692 . . . . . . . 8  |-  ( w  =  y  <->  y  =  w )
2018, 19syl6bb 253 . . . . . . 7  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  y  =  w ) )
2120pm5.32i 619 . . . . . 6  |-  ( ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( z  =  { <. B ,  y
>. }  /\  y  =  w ) )
2221anbi2i 676 . . . . 5  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
23 anass 631 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
24 ancom 438 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
2522, 23, 243bitr2i 265 . . . 4  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
2625exbii 1592 . . 3  |-  ( E. y ( y  e.  A  /\  ( z  =  { <. B , 
y >. }  /\  w  =  ( z `  B ) ) )  <->  E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) ) )
27 vex 2951 . . . 4  |-  w  e. 
_V
28 eleq1 2495 . . . . 5  |-  ( y  =  w  ->  (
y  e.  A  <->  w  e.  A ) )
29 opeq2 3977 . . . . . . 7  |-  ( y  =  w  ->  <. B , 
y >.  =  <. B ,  w >. )
3029sneqd 3819 . . . . . 6  |-  ( y  =  w  ->  { <. B ,  y >. }  =  { <. B ,  w >. } )
3130eqeq2d 2446 . . . . 5  |-  ( y  =  w  ->  (
z  =  { <. B ,  y >. }  <->  z  =  { <. B ,  w >. } ) )
3228, 31anbi12d 692 . . . 4  |-  ( y  =  w  ->  (
( y  e.  A  /\  z  =  { <. B ,  y >. } )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) ) )
3327, 32ceqsexv 2983 . . 3  |-  ( E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
3413, 26, 333bitri 263 . 2  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
351, 2, 4, 6, 34en2i 7136 1  |-  ( A  ^m  { B }
)  ~~  A
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   {csn 3806   <.cop 3809   class class class wbr 4204   ` cfv 5445  (class class class)co 6072    ^m cmap 7009    ~~ cen 7097
This theorem is referenced by:  map2xp  7268  mapdom3  7270  ackbij1lem5  8093  pwxpndom2  8529  hashmap  11686
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-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  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-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-map 7011  df-en 7101
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