Theorem xpsfrnel 16223

Description: Elementhood in the target space of the function F appearing in xpsval 16232. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
xpsfrnel	|- ( G e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )

Distinct variable groups:   A, k   B, k   k, G

Proof of Theorem xpsfrnel

Step	Hyp	Ref	Expression
1		elixp2 7912	. 2 |- ( G e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( G e. _V /\ G Fn 2o /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) )
2		3ancoma 1045	. . 3 |- ( ( G e. _V /\ G Fn 2o /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) <-> ( G Fn 2o /\ G e. _V /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) )
3		df2o3 7573	. . . . . . . 8 |- 2o = { (/) , 1o }
4	3	raleqi 3142	. . . . . . 7 |- ( A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) <-> A. k e. { (/) , 1o } ( G ` k ) e. if ( k = (/) , A , B ) )
5		0ex 4790	. . . . . . . 8 |- (/) e. _V
6		1on 7567	. . . . . . . . 9 |- 1o e. On
7	6	elexi 3213	. . . . . . . 8 |- 1o e. _V
8		fveq2 6191	. . . . . . . . 9 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
9		iftrue 4092	. . . . . . . . 9 |- ( k = (/) -> if ( k = (/) , A , B ) = A )
10	8, 9	eleq12d 2695	. . . . . . . 8 |- ( k = (/) -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` (/) ) e. A ) )
11		fveq2 6191	. . . . . . . . 9 |- ( k = 1o -> ( G ` k ) = ( G ` 1o ) )
12		1n0 7575	. . . . . . . . . . 11 |- 1o =/= (/)
13		neeq1 2856	. . . . . . . . . . 11 |- ( k = 1o -> ( k =/= (/) <-> 1o =/= (/) ) )
14	12, 13	mpbiri 248	. . . . . . . . . 10 |- ( k = 1o -> k =/= (/) )
15		ifnefalse 4098	. . . . . . . . . 10 |- ( k =/= (/) -> if ( k = (/) , A , B ) = B )
16	14, 15	syl 17	. . . . . . . . 9 |- ( k = 1o -> if ( k = (/) , A , B ) = B )
17	11, 16	eleq12d 2695	. . . . . . . 8 |- ( k = 1o -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` 1o ) e. B ) )
18	5, 7, 10, 17	ralpr 4238	. . . . . . 7 |- ( A. k e. { (/) , 1o } ( G ` k ) e. if ( k = (/) , A , B ) <-> ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )
19	4, 18	bitri 264	. . . . . 6 |- ( A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) <-> ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )
20		2onn 7720	. . . . . . . . . 10 |- 2o e. om
21		nnfi 8153	. . . . . . . . . 10 |- ( 2o e. om -> 2o e. Fin )
22	20, 21	ax-mp 5	. . . . . . . . 9 |- 2o e. Fin
23		fnfi 8238	. . . . . . . . 9 |- ( ( G Fn 2o /\ 2o e. Fin ) -> G e. Fin )
24	22, 23	mpan2 707	. . . . . . . 8 |- ( G Fn 2o -> G e. Fin )
25		elex 3212	. . . . . . . 8 |- ( G e. Fin -> G e. _V )
26	24, 25	syl 17	. . . . . . 7 |- ( G Fn 2o -> G e. _V )
27	26	biantrurd 529	. . . . . 6 |- ( G Fn 2o -> ( A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G e. _V /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) ) )
28	19, 27	syl5rbbr 275	. . . . 5 |- ( G Fn 2o -> ( ( G e. _V /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) <-> ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) ) )
29	28	pm5.32i 669	. . . 4 |- ( ( G Fn 2o /\ ( G e. _V /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) ) <-> ( G Fn 2o /\ ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) ) )
30		3anass 1042	. . . 4 |- ( ( G Fn 2o /\ G e. _V /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) <-> ( G Fn 2o /\ ( G e. _V /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) ) )
31		3anass 1042	. . . 4 |- ( ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) <-> ( G Fn 2o /\ ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) ) )
32	29, 30, 31	3bitr4i 292	. . 3 |- ( ( G Fn 2o /\ G e. _V /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) <-> ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )
33	2, 32	bitri 264	. 2 |- ( ( G e. _V /\ G Fn 2o /\ A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) ) <-> ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )
34	1, 33	bitri 264	1 |- ( G e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   ifcif 4086   {cpr 4179   Oncon0 5723   Fn wfn 5883   ` cfv 5888   omcom 7065   1oc1o 7553   2oc2o 7554   X_cixp 7908   Fincfn 7955

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-int 4476  df-iun 4522  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  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-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  xpsfrnel2  16225  xpsff1o  16228