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

Theorem xpsfeq 14836
Description: A function on  2o is determined by its values at zero and one. (Contributed by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
xpsfeq  |-  ( G  Fn  2o  ->  `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } )  =  G )

Proof of Theorem xpsfeq
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fvex 5882 . . . 4  |-  ( G `
 (/) )  e.  _V
2 fvex 5882 . . . 4  |-  ( G `
 1o )  e. 
_V
3 xpscfn 14831 . . . 4  |-  ( ( ( G `  (/) )  e. 
_V  /\  ( G `  1o )  e.  _V )  ->  `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } )  Fn  2o )
41, 2, 3mp2an 672 . . 3  |-  `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } )  Fn  2o
54a1i 11 . 2  |-  ( G  Fn  2o  ->  `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } )  Fn  2o )
6 id 22 . 2  |-  ( G  Fn  2o  ->  G  Fn  2o )
7 elpri 4053 . . . . 5  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
8 df2o3 7155 . . . . 5  |-  2o  =  { (/) ,  1o }
97, 8eleq2s 2575 . . . 4  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
10 xpsc0 14832 . . . . . . 7  |-  ( ( G `  (/) )  e. 
_V  ->  ( `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } ) `  (/) )  =  ( G `  (/) ) )
111, 10ax-mp 5 . . . . . 6  |-  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  (/) )  =  ( G `  (/) )
12 fveq2 5872 . . . . . 6  |-  ( k  =  (/)  ->  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  k
)  =  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  (/) ) )
13 fveq2 5872 . . . . . 6  |-  ( k  =  (/)  ->  ( G `
 k )  =  ( G `  (/) ) )
1411, 12, 133eqtr4a 2534 . . . . 5  |-  ( k  =  (/)  ->  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  k
)  =  ( G `
 k ) )
15 xpsc1 14833 . . . . . . 7  |-  ( ( G `  1o )  e.  _V  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  1o )  =  ( G `  1o ) )
162, 15ax-mp 5 . . . . . 6  |-  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  1o )  =  ( G `  1o )
17 fveq2 5872 . . . . . 6  |-  ( k  =  1o  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } ) `  1o ) )
18 fveq2 5872 . . . . . 6  |-  ( k  =  1o  ->  ( G `  k )  =  ( G `  1o ) )
1916, 17, 183eqtr4a 2534 . . . . 5  |-  ( k  =  1o  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
2014, 19jaoi 379 . . . 4  |-  ( ( k  =  (/)  \/  k  =  1o )  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
219, 20syl 16 . . 3  |-  ( k  e.  2o  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
2221adantl 466 . 2  |-  ( ( G  Fn  2o  /\  k  e.  2o )  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
235, 6, 22eqfnfvd 5985 1  |-  ( G  Fn  2o  ->  `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } )  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)c0 3790   {csn 4033   {cpr 4035   `'ccnv 5004    Fn wfn 5589   ` cfv 5594  (class class class)co 6295   1oc1o 7135   2oc2o 7136    +c ccda 8559
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1o 7142  df-2o 7143  df-cda 8560
This theorem is referenced by:  xpsff1o  14840  xpstopnlem2  20180
  Copyright terms: Public domain W3C validator