Theorem xpsfeq 12928

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 6494	. . . 4 |- (/) e. 2o
2		funfvex 5571	. . . . 5 |- ( ( Fun G /\ (/) e. dom G ) -> ( G ` (/) ) e. _V )
3	2	funfni 5354	. . . 4 |- ( ( G Fn 2o /\ (/) e. 2o ) -> ( G ` (/) ) e. _V )
4	1, 3	mpan2 425	. . 3 |- ( G Fn 2o -> ( G ` (/) ) e. _V )
5		1lt2o 6495	. . . 4 |- 1o e. 2o
6		funfvex 5571	. . . . 5 |- ( ( Fun G /\ 1o e. dom G ) -> ( G ` 1o ) e. _V )
7	6	funfni 5354	. . . 4 |- ( ( G Fn 2o /\ 1o e. 2o ) -> ( G ` 1o ) e. _V )
8	5, 7	mpan2 425	. . 3 |- ( G Fn 2o -> ( G ` 1o ) e. _V )
9		fnpr2o 12922	. . 3 |- ( ( ( G ` (/) ) e. _V /\ ( G ` 1o ) e. _V ) -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } Fn 2o )
10	4, 8, 9	syl2anc 411	. 2 |- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } Fn 2o )
11		id 19	. 2 |- ( G Fn 2o -> G Fn 2o )
12		elpri 3641	. . . 4 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
13		df2o3 6483	. . . 4 |- 2o = { (/) , 1o }
14	12, 13	eleq2s 2288	. . 3 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
15		fvpr0o 12924	. . . . . . 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 5554	. . . . . 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 5554	. . . . . 6 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
21	20	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = (/) ) -> ( G ` k ) = ( G ` (/) ) )
22	17, 19, 21	3eqtr4d 2236	. . . 4 |- ( ( G Fn 2o /\ k = (/) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
23		fvpr1o 12925	. . . . . . 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 5554	. . . . . 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 5554	. . . . . 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 2236	. . . 4 |- ( ( G Fn 2o /\ k = 1o ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
31	22, 30	jaodan 798	. . 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 5658	1 |- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } = G )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2164   _Vcvv 2760   (/)c0 3446   {cpr 3619   <.cop 3621   Fn wfn 5249   ` cfv 5254   1oc1o 6462   2oc2o 6463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-1o 6469  df-2o 6470

This theorem is referenced by:  xpsff1o  12932