Theorem xpsfeq 12818

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 6465	. . . 4 |- (/) e. 2o
2		funfvex 5551	. . . . 5 |- ( ( Fun G /\ (/) e. dom G ) -> ( G ` (/) ) e. _V )
3	2	funfni 5335	. . . 4 |- ( ( G Fn 2o /\ (/) e. 2o ) -> ( G ` (/) ) e. _V )
4	1, 3	mpan2 425	. . 3 |- ( G Fn 2o -> ( G ` (/) ) e. _V )
5		1lt2o 6466	. . . 4 |- 1o e. 2o
6		funfvex 5551	. . . . 5 |- ( ( Fun G /\ 1o e. dom G ) -> ( G ` 1o ) e. _V )
7	6	funfni 5335	. . . 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 12812	. . 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 3630	. . . 4 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
13		df2o3 6454	. . . 4 |- 2o = { (/) , 1o }
14	12, 13	eleq2s 2284	. . 3 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
15		fvpr0o 12814	. . . . . . 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 5534	. . . . . 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 5534	. . . . . 6 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
21	20	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = (/) ) -> ( G ` k ) = ( G ` (/) ) )
22	17, 19, 21	3eqtr4d 2232	. . . 4 |- ( ( G Fn 2o /\ k = (/) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
23		fvpr1o 12815	. . . . . . 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 5534	. . . . . 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 5534	. . . . . 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 2232	. . . 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 5636	1 |- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } = G )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2160   _Vcvv 2752   (/)c0 3437   {cpr 3608   <.cop 3610   Fn wfn 5230   ` cfv 5235   1oc1o 6433   2oc2o 6434

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-1o 6440  df-2o 6441

This theorem is referenced by:  xpsff1o  12822