Theorem xpsfeq 13558

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 6674	. . . 4 |- (/) e. 2o
2		funfvex 5687	. . . . 5 |- ( ( Fun G /\ (/) e. dom G ) -> ( G ` (/) ) e. _V )
3	2	funfni 5458	. . . 4 |- ( ( G Fn 2o /\ (/) e. 2o ) -> ( G ` (/) ) e. _V )
4	1, 3	mpan2 425	. . 3 |- ( G Fn 2o -> ( G ` (/) ) e. _V )
5		1lt2o 6675	. . . 4 |- 1o e. 2o
6		funfvex 5687	. . . . 5 |- ( ( Fun G /\ 1o e. dom G ) -> ( G ` 1o ) e. _V )
7	6	funfni 5458	. . . 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 13552	. . 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 3712	. . . 4 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
13		df2o3 6662	. . . 4 |- 2o = { (/) , 1o }
14	12, 13	eleq2s 2327	. . 3 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
15		fvpr0o 13554	. . . . . . 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 5670	. . . . . 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 5670	. . . . . 6 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
21	20	adantl 277	. . . . 5 |- ( ( G Fn 2o /\ k = (/) ) -> ( G ` k ) = ( G ` (/) ) )
22	17, 19, 21	3eqtr4d 2275	. . . 4 |- ( ( G Fn 2o /\ k = (/) ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
23		fvpr1o 13555	. . . . . . 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 5670	. . . . . 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 5670	. . . . . 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 2275	. . . 4 |- ( ( G Fn 2o /\ k = 1o ) -> ( { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } ` k ) = ( G ` k ) )
31	22, 30	jaodan 805	. . 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 5778	1 |- ( G Fn 2o -> { <. (/) , ( G ` (/) ) >. , <. 1o , ( G ` 1o ) >. } = G )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   _Vcvv 2813   (/)c0 3508   {cpr 3690   <.cop 3692   Fn wfn 5347   ` cfv 5352   1oc1o 6640   2oc2o 6641

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-1o 6647  df-2o 6648

This theorem is referenced by:  xpsff1o  13562