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

Theorem mapsnf1o2 7518
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 5882 . . 3  |-  ( x `
 X )  e. 
_V
2 mapsncnv.f . . 3  |-  F  =  ( x  e.  ( B  ^m  S ) 
|->  ( x `  X
) )
31, 2fnmpti 5715 . 2  |-  F  Fn  ( B  ^m  S )
4 mapsncnv.s . . . . 5  |-  S  =  { X }
5 snex 4654 . . . . 5  |-  { X }  e.  _V
64, 5eqeltri 2504 . . . 4  |-  S  e. 
_V
7 snex 4654 . . . 4  |-  { y }  e.  _V
86, 7xpex 6600 . . 3  |-  ( S  X.  { y } )  e.  _V
9 mapsncnv.b . . . 4  |-  B  e. 
_V
10 mapsncnv.x . . . 4  |-  X  e. 
_V
114, 9, 10, 2mapsncnv 7517 . . 3  |-  `' F  =  ( y  e.  B  |->  ( S  X.  { y } ) )
128, 11fnmpti 5715 . 2  |-  `' F  Fn  B
13 dff1o4 5830 . 2  |-  ( F : ( B  ^m  S ) -1-1-onto-> B  <->  ( F  Fn  ( B  ^m  S )  /\  `' F  Fn  B ) )
143, 12, 13mpbir2an 928 1  |-  F :
( B  ^m  S
)
-1-1-onto-> B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1867   _Vcvv 3078   {csn 3993    |-> cmpt 4475    X. cxp 4843   `'ccnv 4844    Fn wfn 5587   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296    ^m cmap 7471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-map 7473
This theorem is referenced by:  mapsnf1o3  7519  coe1sfi  18747  coe1mul2lem2  18802  ply1coe  18830  ply1coeOLD  18831  evl1var  18865  pf1mpf  18881  pf1ind  18884  deg1ldg  22948  deg1leb  22951
  Copyright terms: Public domain W3C validator