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Theorem mapsnf1o 7105
Description: A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
ixpsnf1o.f  |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )
Assertion
Ref Expression
mapsnf1o  |-  ( ( A  e.  V  /\  I  e.  W )  ->  F : A -1-1-onto-> ( A  ^m  { I }
) )
Distinct variable groups:    x, I    x, A    x, V    x, W
Allowed substitution hint:    F( x)

Proof of Theorem mapsnf1o
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ixpsnf1o.f . . . 4  |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )
21ixpsnf1o 7104 . . 3  |-  ( I  e.  W  ->  F : A -1-1-onto-> X_ y  e.  {
I } A )
32adantl 454 . 2  |-  ( ( A  e.  V  /\  I  e.  W )  ->  F : A -1-1-onto-> X_ y  e.  { I } A
)
4 snex 4407 . . . . 5  |-  { I }  e.  _V
5 ixpconstg 7073 . . . . . 6  |-  ( ( { I }  e.  _V  /\  A  e.  V
)  ->  X_ y  e. 
{ I } A  =  ( A  ^m  { I } ) )
65eqcomd 2443 . . . . 5  |-  ( ( { I }  e.  _V  /\  A  e.  V
)  ->  ( A  ^m  { I } )  =  X_ y  e.  {
I } A )
74, 6mpan 653 . . . 4  |-  ( A  e.  V  ->  ( A  ^m  { I }
)  =  X_ y  e.  { I } A
)
87adantr 453 . . 3  |-  ( ( A  e.  V  /\  I  e.  W )  ->  ( A  ^m  {
I } )  = 
X_ y  e.  {
I } A )
9 f1oeq3 5669 . . 3  |-  ( ( A  ^m  { I } )  =  X_ y  e.  { I } A  ->  ( F : A -1-1-onto-> ( A  ^m  {
I } )  <->  F : A
-1-1-onto-> X_ y  e.  { I } A ) )
108, 9syl 16 . 2  |-  ( ( A  e.  V  /\  I  e.  W )  ->  ( F : A -1-1-onto-> ( A  ^m  { I }
)  <->  F : A -1-1-onto-> X_ y  e.  { I } A
) )
113, 10mpbird 225 1  |-  ( ( A  e.  V  /\  I  e.  W )  ->  F : A -1-1-onto-> ( A  ^m  { I }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    e. cmpt 4268    X. cxp 4878   -1-1-onto->wf1o 5455  (class class class)co 6083    ^m cmap 7020   X_cixp 7065
This theorem is referenced by:  pwssnf1o  13722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-ixp 7066
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