MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mapsnf1o2 Unicode version

Theorem mapsnf1o2 6815
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 5539 . . 3  |-  ( x `
 X )  e. 
_V
2 mapsncnv.f . . 3  |-  F  =  ( x  e.  ( B  ^m  S ) 
|->  ( x `  X
) )
31, 2fnmpti 5372 . 2  |-  F  Fn  ( B  ^m  S )
4 mapsncnv.s . . . . 5  |-  S  =  { X }
5 snex 4216 . . . . 5  |-  { X }  e.  _V
64, 5eqeltri 2353 . . . 4  |-  S  e. 
_V
7 snex 4216 . . . 4  |-  { y }  e.  _V
86, 7xpex 4801 . . 3  |-  ( S  X.  { y } )  e.  _V
9 mapsncnv.b . . . 4  |-  B  e. 
_V
10 mapsncnv.x . . . 4  |-  X  e. 
_V
114, 9, 10, 2mapsncnv 6814 . . 3  |-  `' F  =  ( y  e.  B  |->  ( S  X.  { y } ) )
128, 11fnmpti 5372 . 2  |-  `' F  Fn  B
13 dff1o4 5480 . 2  |-  ( F : ( B  ^m  S ) -1-1-onto-> B  <->  ( F  Fn  ( B  ^m  S )  /\  `' F  Fn  B ) )
143, 12, 13mpbir2an 886 1  |-  F :
( B  ^m  S
)
-1-1-onto-> B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  mapsnf1o3  6816  coe1sfi  16293  coe1mul2lem2  16345  ply1coe  16368  evl1var  19415  pf1mpf  19435  pf1ind  19438  deg1ldg  19478  deg1leb  19481
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774
  Copyright terms: Public domain W3C validator