Theorem restco 12382

Description: Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)

Assertion

Ref	Expression
restco	|- ( ( J e. V /\ A e. W /\ B e. X ) -> ( ( J ↾t A ) ↾t B ) = ( J ↾t ( A i^i B ) ) )

Proof of Theorem restco

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2692	. . . . 5 |- y e. _V
2	1	inex1 4070	. . . 4 |- ( y i^i A ) e. _V
3		ineq1 3275	. . . . 5 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( ( y i^i A ) i^i B ) )
4		inass 3291	. . . . 5 |- ( ( y i^i A ) i^i B ) = ( y i^i ( A i^i B ) )
5	3, 4	eqtrdi 2189	. . . 4 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( y i^i ( A i^i B ) ) )
6	2, 5	abrexco 5668	. . 3 |- { z | E. x e. { w | E. y e. J w = ( y i^i A ) } z = ( x i^i B ) } = { z | E. y e. J z = ( y i^i ( A i^i B ) ) }
7		eqid 2140	. . . . . 6 |- ( y e. J |-> ( y i^i A ) ) = ( y e. J |-> ( y i^i A ) )
8	7	rnmpt 4795	. . . . 5 |- ran ( y e. J |-> ( y i^i A ) ) = { w | E. y e. J w = ( y i^i A ) }
9		mpteq1 4020	. . . . 5 |- ( ran ( y e. J |-> ( y i^i A ) ) = { w | E. y e. J w = ( y i^i A ) } -> ( x e. ran ( y e. J |-> ( y i^i A ) ) |-> ( x i^i B ) ) = ( x e. { w | E. y e. J w = ( y i^i A ) } |-> ( x i^i B ) ) )
10	8, 9	ax-mp 5	. . . 4 |- ( x e. ran ( y e. J |-> ( y i^i A ) ) |-> ( x i^i B ) ) = ( x e. { w | E. y e. J w = ( y i^i A ) } |-> ( x i^i B ) )
11	10	rnmpt 4795	. . 3 |- ran ( x e. ran ( y e. J |-> ( y i^i A ) ) |-> ( x i^i B ) ) = { z | E. x e. { w | E. y e. J w = ( y i^i A ) } z = ( x i^i B ) }
12		eqid 2140	. . . 4 |- ( y e. J |-> ( y i^i ( A i^i B ) ) ) = ( y e. J |-> ( y i^i ( A i^i B ) ) )
13	12	rnmpt 4795	. . 3 |- ran ( y e. J |-> ( y i^i ( A i^i B ) ) ) = { z | E. y e. J z = ( y i^i ( A i^i B ) ) }
14	6, 11, 13	3eqtr4i 2171	. 2 |- ran ( x e. ran ( y e. J |-> ( y i^i A ) ) |-> ( x i^i B ) ) = ran ( y e. J |-> ( y i^i ( A i^i B ) ) )
15		restval 12165	. . . . 5 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
16	15	3adant3 1002	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
17	16	oveq1d 5797	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( ( J ↾t A ) ↾t B ) = ( ran ( y e. J |-> ( y i^i A ) ) ↾t B ) )
18		restfn 12163	. . . . . 6 |- ↾t Fn ( _V X. _V )
19		simp1 982	. . . . . . 7 |- ( ( J e. V /\ A e. W /\ B e. X ) -> J e. V )
20	19	elexd 2702	. . . . . 6 |- ( ( J e. V /\ A e. W /\ B e. X ) -> J e. _V )
21		simp2 983	. . . . . . 7 |- ( ( J e. V /\ A e. W /\ B e. X ) -> A e. W )
22	21	elexd 2702	. . . . . 6 |- ( ( J e. V /\ A e. W /\ B e. X ) -> A e. _V )
23		fnovex 5812	. . . . . 6 |- ( ( ↾t Fn ( _V X. _V ) /\ J e. _V /\ A e. _V ) -> ( J ↾t A ) e. _V )
24	18, 20, 22, 23	mp3an2i 1321	. . . . 5 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t A ) e. _V )
25	16, 24	eqeltrrd 2218	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ran ( y e. J |-> ( y i^i A ) ) e. _V )
26		simp3 984	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> B e. X )
27		restval 12165	. . . 4 |- ( ( ran ( y e. J |-> ( y i^i A ) ) e. _V /\ B e. X ) -> ( ran ( y e. J |-> ( y i^i A ) ) ↾t B ) = ran ( x e. ran ( y e. J |-> ( y i^i A ) ) |-> ( x i^i B ) ) )
28	25, 26, 27	syl2anc 409	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( ran ( y e. J |-> ( y i^i A ) ) ↾t B ) = ran ( x e. ran ( y e. J |-> ( y i^i A ) ) |-> ( x i^i B ) ) )
29	17, 28	eqtrd 2173	. 2 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( ( J ↾t A ) ↾t B ) = ran ( x e. ran ( y e. J |-> ( y i^i A ) ) |-> ( x i^i B ) ) )
30		inex1g 4072	. . . 4 |- ( A e. W -> ( A i^i B ) e. _V )
31	30	3ad2ant2 1004	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( A i^i B ) e. _V )
32		restval 12165	. . 3 |- ( ( J e. V /\ ( A i^i B ) e. _V ) -> ( J ↾t ( A i^i B ) ) = ran ( y e. J |-> ( y i^i ( A i^i B ) ) ) )
33	19, 31, 32	syl2anc 409	. 2 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t ( A i^i B ) ) = ran ( y e. J |-> ( y i^i ( A i^i B ) ) ) )
34	14, 29, 33	3eqtr4a 2199	1 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( ( J ↾t A ) ↾t B ) = ( J ↾t ( A i^i B ) ) )

Syntax hints:   -> wi 4   /\ w3a 963   = wceq 1332   e. wcel 1481   {cab 2126   E.wrex 2418   _Vcvv 2689   i^i cin 3075   |-> cmpt 3997   X. cxp 4545   ran crn 4548   Fn wfn 5126  (class class class)co 5782   ↾_t crest 12159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-rest 12161

This theorem is referenced by:  restabs  12383  restin  12384