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Theorem mapsnf1o2 7467
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 5876 . . 3  |-  ( x `
 X )  e. 
_V
2 mapsncnv.f . . 3  |-  F  =  ( x  e.  ( B  ^m  S ) 
|->  ( x `  X
) )
31, 2fnmpti 5709 . 2  |-  F  Fn  ( B  ^m  S )
4 mapsncnv.s . . . . 5  |-  S  =  { X }
5 snex 4688 . . . . 5  |-  { X }  e.  _V
64, 5eqeltri 2551 . . . 4  |-  S  e. 
_V
7 snex 4688 . . . 4  |-  { y }  e.  _V
86, 7xpex 6589 . . 3  |-  ( S  X.  { y } )  e.  _V
9 mapsncnv.b . . . 4  |-  B  e. 
_V
10 mapsncnv.x . . . 4  |-  X  e. 
_V
114, 9, 10, 2mapsncnv 7466 . . 3  |-  `' F  =  ( y  e.  B  |->  ( S  X.  { y } ) )
128, 11fnmpti 5709 . 2  |-  `' F  Fn  B
13 dff1o4 5824 . 2  |-  ( F : ( B  ^m  S ) -1-1-onto-> B  <->  ( F  Fn  ( B  ^m  S )  /\  `' F  Fn  B ) )
143, 12, 13mpbir2an 918 1  |-  F :
( B  ^m  S
)
-1-1-onto-> B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998    Fn wfn 5583   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6285    ^m cmap 7421
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 6577
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-csb 3436  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-iun 4327  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 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-map 7423
This theorem is referenced by:  mapsnf1o3  7468  coe1sfi  18063  coe1sfiOLD  18064  coe1mul2lem2  18120  ply1coe  18148  ply1coeOLD  18149  evl1var  18183  pf1mpf  18199  pf1ind  18202  deg1ldg  22319  deg1leb  22322
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