Theorem xpsfeq 16224

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 ` (/) ) } +c { ( G ` 1o ) } ) = G )

Proof of Theorem xpsfeq

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 |- ( G ` (/) ) e. _V
2		fvex 6201	. . . 4 |- ( G ` 1o ) e. _V
3		xpscfn 16219	. . . 4 |- ( ( ( G ` (/) ) e. _V /\ ( G ` 1o ) e. _V ) -> `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) Fn 2o )
4	1, 2, 3	mp2an 708	. . 3 |- `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) Fn 2o
5	4	a1i 11	. 2 |- ( G Fn 2o -> `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) Fn 2o )
6		id 22	. 2 |- ( G Fn 2o -> G Fn 2o )
7		elpri 4197	. . . . 5 |- ( k e. { (/) , 1o } -> ( k = (/) \/ k = 1o ) )
8		df2o3 7573	. . . . 5 |- 2o = { (/) , 1o }
9	7, 8	eleq2s 2719	. . . 4 |- ( k e. 2o -> ( k = (/) \/ k = 1o ) )
10		xpsc0 16220	. . . . . . 7 |- ( ( G ` (/) ) e. _V -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` (/) ) = ( G ` (/) ) )
11	1, 10	ax-mp 5	. . . . . 6 |- ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` (/) ) = ( G ` (/) )
12		fveq2 6191	. . . . . 6 |- ( k = (/) -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` k ) = ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` (/) ) )
13		fveq2 6191	. . . . . 6 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
14	11, 12, 13	3eqtr4a 2682	. . . . 5 |- ( k = (/) -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` k ) = ( G ` k ) )
15		xpsc1 16221	. . . . . . 7 |- ( ( G ` 1o ) e. _V -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` 1o ) = ( G ` 1o ) )
16	2, 15	ax-mp 5	. . . . . 6 |- ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` 1o ) = ( G ` 1o )
17		fveq2 6191	. . . . . 6 |- ( k = 1o -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` k ) = ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` 1o ) )
18		fveq2 6191	. . . . . 6 |- ( k = 1o -> ( G ` k ) = ( G ` 1o ) )
19	16, 17, 18	3eqtr4a 2682	. . . . 5 |- ( k = 1o -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` k ) = ( G ` k ) )
20	14, 19	jaoi 394	. . . 4 |- ( ( k = (/) \/ k = 1o ) -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` k ) = ( G ` k ) )
21	9, 20	syl 17	. . 3 |- ( k e. 2o -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` k ) = ( G ` k ) )
22	21	adantl 482	. 2 |- ( ( G Fn 2o /\ k e. 2o ) -> ( `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) ` k ) = ( G ` k ) )
23	5, 6, 22	eqfnfvd 6314	1 |- ( G Fn 2o -> `' ( { ( G ` (/) ) } +c { ( G ` 1o ) } ) = G )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {csn 4177   {cpr 4179   `'ccnv 5113   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1oc1o 7553   2oc2o 7554   +c ccda 8989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-cda 8990

This theorem is referenced by:  xpsff1o  16228  xpstopnlem2  21614