Theorem xpsfrnel 13507

Description: Elementhood in the target space of the function F appearing in xpsval 13515. (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 6914	. 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 6732	. . . . . . . . . 10 |- 2o e. om
4		nnfi 7102	. . . . . . . . . 10 |- ( 2o e. om -> 2o e. Fin )
5	3, 4	ax-mp 5	. . . . . . . . 9 |- 2o e. Fin
6		fnfi 7178	. . . . . . . . 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 2817	. . . . . . 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 6640	. . . . . . . 8 |- 2o = { (/) , 1o }
11	10	raleqi 2735	. . . . . . 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 4221	. . . . . . . 8 |- (/) e. _V
13		1oex 6633	. . . . . . . 8 |- 1o e. _V
14		fveq2 5648	. . . . . . . . 9 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
15		iftrue 3614	. . . . . . . . 9 |- ( k = (/) -> if ( k = (/) , A , B ) = A )
16	14, 15	eleq12d 2302	. . . . . . . 8 |- ( k = (/) -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` (/) ) e. A ) )
17		fveq2 5648	. . . . . . . . 9 |- ( k = 1o -> ( G ` k ) = ( G ` 1o ) )
18		1n0 6643	. . . . . . . . . . 11 |- 1o =/= (/)
19		neeq1 2416	. . . . . . . . . . 11 |- ( k = 1o -> ( k =/= (/) <-> 1o =/= (/) ) )
20	18, 19	mpbiri 168	. . . . . . . . . 10 |- ( k = 1o -> k =/= (/) )
21		ifnefalse 3620	. . . . . . . . . 10 |- ( k =/= (/) -> if ( k = (/) , A , B ) = B )
22	20, 21	syl 14	. . . . . . . . 9 |- ( k = 1o -> if ( k = (/) , A , B ) = B )
23	17, 22	eleq12d 2302	. . . . . . . 8 |- ( k = 1o -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` 1o ) e. B ) )
24	12, 13, 16, 23	ralpr 3728	. . . . . . 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 2202   =/= wne 2403   A.wral 2511   _Vcvv 2803   (/)c0 3496   ifcif 3607   {cpr 3674   omcom 4694   Fn wfn 5328   ` cfv 5333   1oc1o 6618   2oc2o 6619   X_cixp 6910   Fincfn 6952

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-2o 6626  df-er 6745  df-ixp 6911  df-en 6953  df-fin 6955

This theorem is referenced by:  xpsfrnel2  13509  xpsff1o  13512