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Theorem xpsfeq 14514
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 5713 . . . 4  |-  ( G `
 (/) )  e.  _V
2 fvex 5713 . . . 4  |-  ( G `
 1o )  e. 
_V
3 xpscfn 14509 . . . 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 3909 . . . . 5  |-  ( k  e.  { (/) ,  1o }  ->  ( k  =  (/)  \/  k  =  1o ) )
8 df2o3 6945 . . . . 5  |-  2o  =  { (/) ,  1o }
97, 8eleq2s 2535 . . . 4  |-  ( k  e.  2o  ->  (
k  =  (/)  \/  k  =  1o ) )
10 xpsc0 14510 . . . . . . 7  |-  ( ( G `  (/) )  e. 
_V  ->  ( `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } ) `  (/) )  =  ( G `  (/) ) )
111, 10ax-mp 5 . . . . . 6  |-  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  (/) )  =  ( G `  (/) )
12 fveq2 5703 . . . . . 6  |-  ( k  =  (/)  ->  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  k
)  =  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  (/) ) )
13 fveq2 5703 . . . . . 6  |-  ( k  =  (/)  ->  ( G `
 k )  =  ( G `  (/) ) )
1411, 12, 133eqtr4a 2501 . . . . 5  |-  ( k  =  (/)  ->  ( `' ( { ( G `
 (/) ) }  +c  { ( G `  1o ) } ) `  k
)  =  ( G `
 k ) )
15 xpsc1 14511 . . . . . . 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 5703 . . . . . 6  |-  ( k  =  1o  ->  ( `' ( { ( G `  (/) ) }  +c  { ( G `
 1o ) } ) `  k )  =  ( `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } ) `  1o ) )
18 fveq2 5703 . . . . . 6  |-  ( k  =  1o  ->  ( G `  k )  =  ( G `  1o ) )
1916, 17, 183eqtr4a 2501 . . . . 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 5812 1  |-  ( G  Fn  2o  ->  `' ( { ( G `  (/) ) }  +c  {
( G `  1o ) } )  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1369    e. wcel 1756   _Vcvv 2984   (/)c0 3649   {csn 3889   {cpr 3891   `'ccnv 4851    Fn wfn 5425   ` cfv 5430  (class class class)co 6103   1oc1o 6925   2oc2o 6926    +c ccda 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1o 6932  df-2o 6933  df-cda 8349
This theorem is referenced by:  xpsff1o  14518  xpstopnlem2  19396
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