Theorem xpsfrnel 13608

Description: Elementhood in the target space of the function F appearing in xpsval 14143. (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 6950	. 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 1012	. . 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		2onn 6767	. . . . . . . . . 10 |- 2o e. om
4		nnfi 7140	. . . . . . . . . 10 |- ( 2o e. om -> 2o e. Fin )
5	3, 4	ax-mp 5	. . . . . . . . 9 |- 2o e. Fin
6		fnfi 7216	. . . . . . . . 9 |- ( ( G Fn 2o /\ 2o e. Fin ) -> G e. Fin )
7	5, 6	mpan2 425	. . . . . . . 8 |- ( G Fn 2o -> G e. Fin )
8	7	elexd 2829	. . . . . . 7 |- ( G Fn 2o -> G e. _V )
9	8	biantrurd 305	. . . . . 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 ) ) ) )
10		df2o3 6675	. . . . . . . 8 |- 2o = { (/) , 1o }
11	10	raleqi 2747	. . . . . . 7 |- ( A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) <-> A. k e. { (/) , 1o } ( G ` k ) e. if ( k = (/) , A , B ) )
12		0ex 4242	. . . . . . . 8 |- (/) e. _V
13		1oex 6668	. . . . . . . 8 |- 1o e. _V
14		fveq2 5675	. . . . . . . . 9 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
15		iftrue 3631	. . . . . . . . 9 |- ( k = (/) -> if ( k = (/) , A , B ) = A )
16	14, 15	eleq12d 2305	. . . . . . . 8 |- ( k = (/) -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` (/) ) e. A ) )
17		fveq2 5675	. . . . . . . . 9 |- ( k = 1o -> ( G ` k ) = ( G ` 1o ) )
18		1n0 6678	. . . . . . . . . . 11 |- 1o =/= (/)
19		neeq1 2427	. . . . . . . . . . 11 |- ( k = 1o -> ( k =/= (/) <-> 1o =/= (/) ) )
20	18, 19	mpbiri 168	. . . . . . . . . 10 |- ( k = 1o -> k =/= (/) )
21		ifnefalse 3637	. . . . . . . . . 10 |- ( k =/= (/) -> if ( k = (/) , A , B ) = B )
22	20, 21	syl 14	. . . . . . . . 9 |- ( k = 1o -> if ( k = (/) , A , B ) = B )
23	17, 22	eleq12d 2305	. . . . . . . 8 |- ( k = 1o -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` 1o ) e. B ) )
24	12, 13, 16, 23	ralpr 3749	. . . . . . 7 |- ( A. k e. { (/) , 1o } ( G ` k ) e. if ( k = (/) , A , B ) <-> ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )
25	11, 24	bitri 184	. . . . . 6 |- ( A. k e. 2o ( G ` k ) e. if ( k = (/) , A , B ) <-> ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )
26	9, 25	bitr3di 195	. . . . 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 ) ) )
27	26	pm5.32i 454	. . . 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 ) ) )
28		3anass 1009	. . . 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 ) ) ) )
29		3anass 1009	. . . 4 |- ( ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) <-> ( G Fn 2o /\ ( ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) ) )
30	27, 28, 29	3bitr4i 212	. . 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 ) )
31	2, 30	bitri 184	. 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 ) )
32	1, 31	bitri 184	1 |- ( G e. X_ k e. 2o if ( k = (/) , A , B ) <-> ( G Fn 2o /\ ( G ` (/) ) e. A /\ ( G ` 1o ) e. B ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   _Vcvv 2815   (/)c0 3512   ifcif 3624   {cpr 3695   omcom 4717   Fn wfn 5352   ` cfv 5357   1oc1o 6653   2oc2o 6654   X_cixp 6946   Fincfn 6988

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1o 6660  df-2o 6661  df-er 6780  df-ixp 6947  df-en 6989  df-fin 6991

This theorem is referenced by:  xpsfrnel2  13610  xpsff1o  13613