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Theorem mapsnen 6698
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 fnmap 6542 . . 3  |-  ^m  Fn  ( _V  X.  _V )
2 mapsnen.1 . . 3  |-  A  e. 
_V
3 mapsnen.2 . . . 4  |-  B  e. 
_V
43snex 4104 . . 3  |-  { B }  e.  _V
5 fnovex 5797 . . 3  |-  ( (  ^m  Fn  ( _V 
X.  _V )  /\  A  e.  _V  /\  { B }  e.  _V )  ->  ( A  ^m  { B } )  e.  _V )
61, 2, 4, 5mp3an 1315 . 2  |-  ( A  ^m  { B }
)  e.  _V
7 vex 2684 . . . 4  |-  z  e. 
_V
87, 3fvex 5434 . . 3  |-  ( z `
 B )  e. 
_V
98a1i 9 . 2  |-  ( z  e.  ( A  ^m  { B } )  -> 
( z `  B
)  e.  _V )
10 vex 2684 . . . . 5  |-  w  e. 
_V
113, 10opex 4146 . . . 4  |-  <. B ,  w >.  e.  _V
1211snex 4104 . . 3  |-  { <. B ,  w >. }  e.  _V
1312a1i 9 . 2  |-  ( w  e.  A  ->  { <. B ,  w >. }  e.  _V )
142, 3mapsn 6577 . . . . . 6  |-  ( A  ^m  { B }
)  =  { z  |  E. y  e.  A  z  =  { <. B ,  y >. } }
1514abeq2i 2248 . . . . 5  |-  ( z  e.  ( A  ^m  { B } )  <->  E. y  e.  A  z  =  { <. B ,  y
>. } )
1615anbi1i 453 . . . 4  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
17 r19.41v 2585 . . . 4  |-  ( E. y  e.  A  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( E. y  e.  A  z  =  { <. B ,  y
>. }  /\  w  =  ( z `  B
) ) )
18 df-rex 2420 . . . 4  |-  ( E. y  e.  A  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  E. y
( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) ) )
1916, 17, 183bitr2i 207 . . 3  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  E. y
( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) ) )
20 fveq1 5413 . . . . . . . . . 10  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  ( { <. B ,  y >. } `  B ) )
21 vex 2684 . . . . . . . . . . 11  |-  y  e. 
_V
223, 21fvsn 5608 . . . . . . . . . 10  |-  ( {
<. B ,  y >. } `  B )  =  y
2320, 22syl6eq 2186 . . . . . . . . 9  |-  ( z  =  { <. B , 
y >. }  ->  (
z `  B )  =  y )
2423eqeq2d 2149 . . . . . . . 8  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  w  =  y ) )
25 equcom 1682 . . . . . . . 8  |-  ( w  =  y  <->  y  =  w )
2624, 25syl6bb 195 . . . . . . 7  |-  ( z  =  { <. B , 
y >. }  ->  (
w  =  ( z `
 B )  <->  y  =  w ) )
2726pm5.32i 449 . . . . . 6  |-  ( ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) )  <->  ( z  =  { <. B ,  y
>. }  /\  y  =  w ) )
2827anbi2i 452 . . . . 5  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
29 anass 398 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  e.  A  /\  (
z  =  { <. B ,  y >. }  /\  y  =  w )
) )
30 ancom 264 . . . . 5  |-  ( ( ( y  e.  A  /\  z  =  { <. B ,  y >. } )  /\  y  =  w )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
3128, 29, 303bitr2i 207 . . . 4  |-  ( ( y  e.  A  /\  ( z  =  { <. B ,  y >. }  /\  w  =  ( z `  B ) ) )  <->  ( y  =  w  /\  (
y  e.  A  /\  z  =  { <. B , 
y >. } ) ) )
3231exbii 1584 . . 3  |-  ( E. y ( y  e.  A  /\  ( z  =  { <. B , 
y >. }  /\  w  =  ( z `  B ) ) )  <->  E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) ) )
33 eleq1w 2198 . . . . 5  |-  ( y  =  w  ->  (
y  e.  A  <->  w  e.  A ) )
34 opeq2 3701 . . . . . . 7  |-  ( y  =  w  ->  <. B , 
y >.  =  <. B ,  w >. )
3534sneqd 3535 . . . . . 6  |-  ( y  =  w  ->  { <. B ,  y >. }  =  { <. B ,  w >. } )
3635eqeq2d 2149 . . . . 5  |-  ( y  =  w  ->  (
z  =  { <. B ,  y >. }  <->  z  =  { <. B ,  w >. } ) )
3733, 36anbi12d 464 . . . 4  |-  ( y  =  w  ->  (
( y  e.  A  /\  z  =  { <. B ,  y >. } )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) ) )
3810, 37ceqsexv 2720 . . 3  |-  ( E. y ( y  =  w  /\  ( y  e.  A  /\  z  =  { <. B ,  y
>. } ) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
3919, 32, 383bitri 205 . 2  |-  ( ( z  e.  ( A  ^m  { B }
)  /\  w  =  ( z `  B
) )  <->  ( w  e.  A  /\  z  =  { <. B ,  w >. } ) )
406, 2, 9, 13, 39en2i 6657 1  |-  ( A  ^m  { B }
)  ~~  A
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2415   _Vcvv 2681   {csn 3522   <.cop 3525   class class class wbr 3924    X. cxp 4532    Fn wfn 5113   ` cfv 5118  (class class class)co 5767    ^m cmap 6535    ~~ cen 6625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-map 6537  df-en 6628
This theorem is referenced by: (None)
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