Theorem xpsfeq 13666

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 ` (/) ) >. , <. 1o , ( G ` 1o ) >. } = G )

Proof of Theorem xpsfeq

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0lt2o 6714	. . . 4 |- (/) e. 2o
2		funfvex 5712	. . . . 5 |- ( ( Fun G /\ (/) e. dom G ) -> ( G ` (/) ) e. _V )
3	2	funfni 5483	. . . 4 |- ( ( G Fn 2o /\ (/) e. 2o ) -> ( G ` (/) ) e. _V )
4	1, 3	mpan2 429	. . 3 |- ( G Fn 2o -> ( G ` (/) ) e. _V )
5		1lt2o 6715	. . . 4 |- 1o e. 2o
6		funfvex 5712	. . . . 5 |- ( ( Fun G /\ 1o e. dom G ) -> ( G ` 1o ) e. _V )
7	6	funfni 5483	. . . 4 |- ( ( G Fn 2o /\ 1o e. 2o ) -> ( G ` 1o ) e. _V )
8	5, 7	mpan2 429	. . 3 |- ( G Fn 2o -> ( G ` 1o ) e. _V )
9		fnpr2o 13660	. . 3 |- ( ( ( G ` (/) ) e. _V /\ ( G ` 1o ) e. _V ) -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } Fn 2o )
10	4, 8, 9	syl2anc 415	. 2 |- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } Fn 2o )
11		id 19	. 2 |- ( G Fn 2o -> G Fn 2o )
12		elpri 3732	. . . 4 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
13		df2o3 6702	. . . 4 |- 2o = { (/) , 1o }
14	12, 13	eleq2s 2333	. . 3 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
15		fvpr0o 13662	. . . . . . 7 |- ( ( G ` (/) ) e. _V -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` (/) ) = ( G ` (/) ) )
16	4, 15	syl 14	. . . . . 6 |- ( G Fn 2o -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` (/) ) = ( G ` (/) ) )
17	16	adantr 276	. . . . 5 |- ( ( G Fn 2o /\ k = (/) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` (/) ) = ( G ` (/) ) )
18		fveq2 5695	. . . . . 6 |- ( k = (/) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` (/) ) )
19	18	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = (/) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` (/) ) )
20		fveq2 5695	. . . . . 6 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
21	20	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = (/) ) -> ( G ` k ) = ( G ` (/) ) )
22	17, 19, 21	3eqtr4d 2281	. . . 4 |- ( ( G Fn 2o /\ k = (/) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
23		fvpr1o 13663	. . . . . . 7 |- ( ( G ` 1o ) e. _V -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` 1o ) = ( G ` 1o ) )
24	8, 23	syl 14	. . . . . 6 |- ( G Fn 2o -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` 1o ) = ( G ` 1o ) )
25	24	adantr 276	. . . . 5 |- ( ( G Fn 2o /\ k = 1o ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` 1o ) = ( G ` 1o ) )
26		fveq2 5695	. . . . . 6 |- ( k = 1o -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` 1o ) )
27	26	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = 1o ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` 1o ) )
28		fveq2 5695	. . . . . 6 |- ( k = 1o -> ( G ` k ) = ( G ` 1o ) )
29	28	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = 1o ) -> ( G ` k ) = ( G ` 1o ) )
30	25, 27, 29	3eqtr4d 2281	. . . 4 |- ( ( G Fn 2o /\ k = 1o ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
31	22, 30	jaodan 809	. . 3 |- ( ( G Fn 2o /\ ( k = (/) \/ k = 1o ) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
32	14, 31	sylan2 286	. 2 |- ( ( G Fn 2o /\ k e. 2o ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
33	10, 11, 32	eqfnfvd 5809	1 |- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } = G )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {cpr 3710   <.cop 3712   Fn wfn 5372   ` cfv 5377   1oc1o 6680   2oc2o 6681

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-1o 6687  df-2o 6688

This theorem is used by:  xpsff1o  13670