Theorem xpsfrnel 13209

Description: Elementhood in the target space of the function F appearing in xpsval 13217. (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 6791	. 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 988	. . 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 6609	. . . . . . . . . 10 |- 2o e. om
4		nnfi 6971	. . . . . . . . . 10 |- ( 2o e. om -> 2o e. Fin )
5	3, 4	ax-mp 5	. . . . . . . . 9 |- 2o e. Fin
6		fnfi 7040	. . . . . . . . 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 2785	. . . . . . 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 6518	. . . . . . . 8 |- 2o = { (/) , 1o }
11	10	raleqi 2706	. . . . . . 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 4172	. . . . . . . 8 |- (/) e. _V
13		1oex 6512	. . . . . . . 8 |- 1o e. _V
14		fveq2 5578	. . . . . . . . 9 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
15		iftrue 3576	. . . . . . . . 9 |- ( k = (/) -> if ( k = (/) , A , B ) = A )
16	14, 15	eleq12d 2276	. . . . . . . 8 |- ( k = (/) -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` (/) ) e. A ) )
17		fveq2 5578	. . . . . . . . 9 |- ( k = 1o -> ( G ` k ) = ( G ` 1o ) )
18		1n0 6520	. . . . . . . . . . 11 |- 1o =/= (/)
19		neeq1 2389	. . . . . . . . . . 11 |- ( k = 1o -> ( k =/= (/) <-> 1o =/= (/) ) )
20	18, 19	mpbiri 168	. . . . . . . . . 10 |- ( k = 1o -> k =/= (/) )
21		ifnefalse 3582	. . . . . . . . . 10 |- ( k =/= (/) -> if ( k = (/) , A , B ) = B )
22	20, 21	syl 14	. . . . . . . . 9 |- ( k = 1o -> if ( k = (/) , A , B ) = B )
23	17, 22	eleq12d 2276	. . . . . . . 8 |- ( k = 1o -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` 1o ) e. B ) )
24	12, 13, 16, 23	ralpr 3688	. . . . . . 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 985	. . . 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 985	. . . 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 981   = wceq 1373   e. wcel 2176   =/= wne 2376   A.wral 2484   _Vcvv 2772   (/)c0 3460   ifcif 3571   {cpr 3634   omcom 4639   Fn wfn 5267   ` cfv 5272   1oc1o 6497   2oc2o 6498   X_cixp 6787   Fincfn 6829

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-1o 6504  df-2o 6505  df-er 6622  df-ixp 6788  df-en 6830  df-fin 6832

This theorem is referenced by:  xpsfrnel2  13211  xpsff1o  13214