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Theorem mapsnf1o 7503
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 7502 . . 3  |-  ( I  e.  W  ->  F : A -1-1-onto-> X_ y  e.  {
I } A )
32adantl 464 . 2  |-  ( ( A  e.  V  /\  I  e.  W )  ->  F : A -1-1-onto-> X_ y  e.  { I } A
)
4 snex 4678 . . . . 5  |-  { I }  e.  _V
5 ixpconstg 7471 . . . . . 6  |-  ( ( { I }  e.  _V  /\  A  e.  V
)  ->  X_ y  e. 
{ I } A  =  ( A  ^m  { I } ) )
65eqcomd 2462 . . . . 5  |-  ( ( { I }  e.  _V  /\  A  e.  V
)  ->  ( A  ^m  { I } )  =  X_ y  e.  {
I } A )
74, 6mpan 668 . . . 4  |-  ( A  e.  V  ->  ( A  ^m  { I }
)  =  X_ y  e.  { I } A
)
87adantr 463 . . 3  |-  ( ( A  e.  V  /\  I  e.  W )  ->  ( A  ^m  {
I } )  = 
X_ y  e.  {
I } A )
9 f1oeq3 5791 . . 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 232 1  |-  ( ( A  e.  V  /\  I  e.  W )  ->  F : A -1-1-onto-> ( A  ^m  { I }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016    |-> cmpt 4497    X. cxp 4986   -1-1-onto->wf1o 5569  (class class class)co 6270    ^m cmap 7412   X_cixp 7462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-ixp 7463
This theorem is referenced by:  pwssnf1o  14987  mat1f1o  19147
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