Theorem restco 17182

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 2919	. . . . 5 |- y e. _V
2	1	inex1 4304	. . . 4 |- ( y i^i A ) e. _V
3		ineq1 3495	. . . . 5 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( ( y i^i A ) i^i B ) )
4		inass 3511	. . . . 5 |- ( ( y i^i A ) i^i B ) = ( y i^i ( A i^i B ) )
5	3, 4	syl6eq 2452	. . . 4 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( y i^i ( A i^i B ) ) )
6	2, 5	abrexco 5945	. . 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 2404	. . . . . 6 |- ( y e. J |-> ( y i^i A ) ) = ( y e. J |-> ( y i^i A ) )
8	7	rnmpt 5075	. . . . 5 |- ran ( y e. J |-> ( y i^i A ) ) = { w | E. y e. J w = ( y i^i A ) }
9		mpteq1 4249	. . . . 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 8	. . . 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 5075	. . 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 2404	. . . 4 |- ( y e. J |-> ( y i^i ( A i^i B ) ) ) = ( y e. J |-> ( y i^i ( A i^i B ) ) )
13	12	rnmpt 5075	. . 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 2434	. 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 13609	. . . . 5 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
16	15	3adant3 977	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
17	16	oveq1d 6055	. . 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		ovex 6065	. . . . 5 |- ( J ↾t A ) e. _V
19	16, 18	syl6eqelr 2493	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ran ( y e. J |-> ( y i^i A ) ) e. _V )
20		simp3 959	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> B e. X )
21		restval 13609	. . . 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 ) ) )
22	19, 20, 21	syl2anc 643	. . 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 ) ) )
23	17, 22	eqtrd 2436	. 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 ) ) )
24		simp1 957	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> J e. V )
25		inex1g 4306	. . . 4 |- ( A e. W -> ( A i^i B ) e. _V )
26	25	3ad2ant2 979	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( A i^i B ) e. _V )
27		restval 13609	. . 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 ) ) ) )
28	24, 26, 27	syl2anc 643	. 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 ) ) ) )
29	14, 23, 28	3eqtr4a 2462	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 936   = wceq 1649   e. wcel 1721   {cab 2390   E.wrex 2667   _Vcvv 2916   i^i cin 3279   e. cmpt 4226   ran crn 4838  (class class class)co 6040   ↾_t crest 13603

This theorem is referenced by:  restabs  17183  restin  17184  resstopn  17204  ressuss  18246

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-rest 13605