Theorem xpsfeq 13418

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 6604	. . . 4 |- (/) e. 2o
2		funfvex 5652	. . . . 5 |- ( ( Fun G /\ (/) e. dom G ) -> ( G ` (/) ) e. _V )
3	2	funfni 5429	. . . 4 |- ( ( G Fn 2o /\ (/) e. 2o ) -> ( G ` (/) ) e. _V )
4	1, 3	mpan2 425	. . 3 |- ( G Fn 2o -> ( G ` (/) ) e. _V )
5		1lt2o 6605	. . . 4 |- 1o e. 2o
6		funfvex 5652	. . . . 5 |- ( ( Fun G /\ 1o e. dom G ) -> ( G ` 1o ) e. _V )
7	6	funfni 5429	. . . 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 13412	. . 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 3690	. . . 4 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
13		df2o3 6592	. . . 4 |- 2o = { (/) , 1o }
14	12, 13	eleq2s 2324	. . 3 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
15		fvpr0o 13414	. . . . . . 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 5635	. . . . . 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 5635	. . . . . 6 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
21	20	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = (/) ) -> ( G ` k ) = ( G ` (/) ) )
22	17, 19, 21	3eqtr4d 2272	. . . 4 |- ( ( G Fn 2o /\ k = (/) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
23		fvpr1o 13415	. . . . . . 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 5635	. . . . . 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 5635	. . . . . 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 2272	. . . 4 |- ( ( G Fn 2o /\ k = 1o ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
31	22, 30	jaodan 802	. . 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 5743	1 |- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } = G )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   _Vcvv 2800   (/)c0 3492   {cpr 3668   <.cop 3670   Fn wfn 5319   ` cfv 5324   1oc1o 6570   2oc2o 6571

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-1o 6577  df-2o 6578

This theorem is referenced by:  xpsff1o  13422