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Theorem xpsfeq 15548
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 5889 . . . 4  |-  ( G `
 (/) )  e.  _V
2 fvex 5889 . . . 4  |-  ( G `
 1o )  e. 
_V
3 xpscfn 15543 . . . 4  |-  ( ( ( G `  (/) )  e. 
_V  /\  ( G `  1o )  e.  _V )  ->  `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } )  Fn  2o )
41, 2, 3mp2an 686 . . 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 3976 . . . . 5  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
8 df2o3 7213 . . . . 5  |-  2o  =  { (/) ,  1o }
97, 8eleq2s 2567 . . . 4  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
10 xpsc0 15544 . . . . . . 7  |-  ( ( G `  (/) )  e. 
_V  ->  ( `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } ) `  (/) )  =  ( G `  (/) ) )
111, 10ax-mp 5 . . . . . 6  |-  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  (/) )  =  ( G `  (/) )
12 fveq2 5879 . . . . . 6  |-  ( k  =  (/)  ->  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  k
)  =  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  (/) ) )
13 fveq2 5879 . . . . . 6  |-  ( k  =  (/)  ->  ( G `
 k )  =  ( G `  (/) ) )
1411, 12, 133eqtr4a 2531 . . . . 5  |-  ( k  =  (/)  ->  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  k
)  =  ( G `
 k ) )
15 xpsc1 15545 . . . . . . 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 5879 . . . . . 6  |-  ( k  =  1o  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } ) `  1o ) )
18 fveq2 5879 . . . . . 6  |-  ( k  =  1o  ->  ( G `  k )  =  ( G `  1o ) )
1916, 17, 183eqtr4a 2531 . . . . 5  |-  ( k  =  1o  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
2014, 19jaoi 386 . . . 4  |-  ( ( k  =  (/)  \/  k  =  1o )  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
219, 20syl 17 . . 3  |-  ( k  e.  2o  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
2221adantl 473 . 2  |-  ( ( G  Fn  2o  /\  k  e.  2o )  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( G `  k ) )
235, 6, 22eqfnfvd 5994 1  |-  ( G  Fn  2o  ->  `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } )  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 375    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   {csn 3959   {cpr 3961   `'ccnv 4838    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   1oc1o 7193   2oc2o 7194    +c ccda 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1o 7200  df-2o 7201  df-cda 8616
This theorem is referenced by:  xpsff1o  15552  xpstopnlem2  20903
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