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Theorem mapsnen 7593
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 6309 . 2  |-  ( A  ^m  { B }
)  e.  _V
2 mapsnen.1 . 2  |-  A  e. 
_V
3 fvex 5876 . . 3  |-  ( z `
 B )  e. 
_V
43a1i 11 . 2  |-  ( z  e.  ( A  ^m  { B } )  -> 
( z `  B
)  e.  _V )
5 snex 4688 . . 3  |-  { <. B ,  w >. }  e.  _V
65a1i 11 . 2  |-  ( w  e.  A  ->  { <. B ,  w >. }  e.  _V )
7 mapsnen.2 . . . . . . 7  |-  B  e. 
_V
82, 7mapsn 7460 . . . . . 6  |-  ( A  ^m  { B }
)  =  { z  |  E. y  e.  A  z  =  { <. B ,  y >. } }
98abeq2i 2594 . . . . 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 3014 . . . 4  |-  ( E. y  e.  A  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
12 df-rex 2820 . . . 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 5865 . . . . . . . . . 10  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  ( { <. B ,  y >. } `  B ) )
15 vex 3116 . . . . . . . . . . 11  |-  y  e. 
_V
167, 15fvsn 6094 . . . . . . . . . 10  |-  ( {
<. B ,  y >. } `  B )  =  y
1714, 16syl6eq 2524 . . . . . . . . 9  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  y )
1817eqeq2d 2481 . . . . . . . 8  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  w  =  y ) )
19 equcom 1743 . . . . . . . 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 1644 . . 3  |-  ( E. y ( y  e.  A  /\  ( z  =  { <. B , 
y >. }  /\  w  =  ( z `  B ) ) )  <->  E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) ) )
27 vex 3116 . . . 4  |-  w  e. 
_V
28 eleq1 2539 . . . . 5  |-  ( y  =  w  ->  (
y  e.  A  <->  w  e.  A ) )
29 opeq2 4214 . . . . . . 7  |-  ( y  =  w  ->  <. B , 
y >.  =  <. B ,  w >. )
3029sneqd 4039 . . . . . 6  |-  ( y  =  w  ->  { <. B ,  y >. }  =  { <. B ,  w >. } )
3130eqeq2d 2481 . . . . 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 3150 . . 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 7553 1  |-  ( A  ^m  { B }
)  ~~  A
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   _Vcvv 3113   {csn 4027   <.cop 4033   class class class wbr 4447   ` cfv 5588  (class class class)co 6284    ^m cmap 7420    ~~ cen 7513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-en 7517
This theorem is referenced by:  map2xp  7687  mapdom3  7689  ackbij1lem5  8604  pwxpndom2  9043  hashmap  12459
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