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Theorem mapsnf1o2 6997
Description: Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)
Hypotheses
Ref Expression
mapsncnv.s  |-  S  =  { X }
mapsncnv.b  |-  B  e. 
_V
mapsncnv.x  |-  X  e. 
_V
mapsncnv.f  |-  F  =  ( x  e.  ( B  ^m  S ) 
|->  ( x `  X
) )
Assertion
Ref Expression
mapsnf1o2  |-  F :
( B  ^m  S
)
-1-1-onto-> B
Distinct variable groups:    x, B    x, S
Allowed substitution hints:    F( x)    X( x)

Proof of Theorem mapsnf1o2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fvex 5682 . . 3  |-  ( x `
 X )  e. 
_V
2 mapsncnv.f . . 3  |-  F  =  ( x  e.  ( B  ^m  S ) 
|->  ( x `  X
) )
31, 2fnmpti 5513 . 2  |-  F  Fn  ( B  ^m  S )
4 mapsncnv.s . . . . 5  |-  S  =  { X }
5 snex 4346 . . . . 5  |-  { X }  e.  _V
64, 5eqeltri 2457 . . . 4  |-  S  e. 
_V
7 snex 4346 . . . 4  |-  { y }  e.  _V
86, 7xpex 4930 . . 3  |-  ( S  X.  { y } )  e.  _V
9 mapsncnv.b . . . 4  |-  B  e. 
_V
10 mapsncnv.x . . . 4  |-  X  e. 
_V
114, 9, 10, 2mapsncnv 6996 . . 3  |-  `' F  =  ( y  e.  B  |->  ( S  X.  { y } ) )
128, 11fnmpti 5513 . 2  |-  `' F  Fn  B
13 dff1o4 5622 . 2  |-  ( F : ( B  ^m  S ) -1-1-onto-> B  <->  ( F  Fn  ( B  ^m  S )  /\  `' F  Fn  B ) )
143, 12, 13mpbir2an 887 1  |-  F :
( B  ^m  S
)
-1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2899   {csn 3757    e. cmpt 4207    X. cxp 4816   `'ccnv 4817    Fn wfn 5389   -1-1-onto->wf1o 5393   ` cfv 5394  (class class class)co 6020    ^m cmap 6954
This theorem is referenced by:  mapsnf1o3  6998  coe1sfi  16537  coe1mul2lem2  16588  ply1coe  16611  evl1var  19819  pf1mpf  19839  pf1ind  19842  deg1ldg  19882  deg1leb  19885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-map 6956
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