Theorem xpsfrnel 13724

Description: Elementhood in the target space of the function F appearing in xpsval 14308. (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 6988	. 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 1016	. . 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 6798	. . . . . . . . . 10 |- 2o e. om
4		nnfi 7178	. . . . . . . . . 10 |- ( 2o e. om -> 2o e. Fin )
5	3, 4	ax-mp 5	. . . . . . . . 9 |- 2o e. Fin
6		fnfi 7254	. . . . . . . . 9 |- ( ( G Fn 2o /\ 2o e. Fin ) -> G e. Fin )
7	5, 6	mpan2 429	. . . . . . . 8 |- ( G Fn 2o -> G e. Fin )
8	7	elexd 2835	. . . . . . 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 6706	. . . . . . . 8 |- 2o = { (/) , 1o }
11	10	raleqi 2753	. . . . . . 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 4262	. . . . . . . 8 |- (/) e. _V
13		1oex 6699	. . . . . . . 8 |- 1o e. _V
14		fveq2 5698	. . . . . . . . 9 |- ( k = (/) -> ( G ` k ) = ( G ` (/) ) )
15		iftrue 3645	. . . . . . . . 9 |- ( k = (/) -> if ( k = (/) , A , B ) = A )
16	14, 15	eleq12d 2309	. . . . . . . 8 |- ( k = (/) -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` (/) ) e. A ) )
17		fveq2 5698	. . . . . . . . 9 |- ( k = 1o -> ( G ` k ) = ( G ` 1o ) )
18		1n0 6709	. . . . . . . . . . 11 |- 1o =/= (/)
19		neeq1 2433	. . . . . . . . . . 11 |- ( k = 1o -> ( k =/= (/) <-> 1o =/= (/) ) )
20	18, 19	mpbiri 168	. . . . . . . . . 10 |- ( k = 1o -> k =/= (/) )
21		ifnefalse 3651	. . . . . . . . . 10 |- ( k =/= (/) -> if ( k = (/) , A , B ) = B )
22	20, 21	syl 14	. . . . . . . . 9 |- ( k = 1o -> if ( k = (/) , A , B ) = B )
23	17, 22	eleq12d 2309	. . . . . . . 8 |- ( k = 1o -> ( ( G ` k ) e. if ( k = (/) , A , B ) <-> ( G ` 1o ) e. B ) )
24	12, 13, 16, 23	ralpr 3764	. . . . . . 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 458	. . . 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 1013	. . . 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 1013	. . . 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 ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   _Vcvv 2821   (/)c0 3520   ifcif 3638   {cpr 3710   omcom 4739   Fn wfn 5375   ` cfv 5380   1oc1o 6684   2oc2o 6685   X_cixp 6984   Fincfn 7026

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1o 6691  df-2o 6692  df-er 6811  df-ixp 6985  df-en 7027  df-fin 7029

This theorem is used by:  xpsfrnel2  13726  xpsff1o  13729