Theorem restco 19533

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 3121	. . . . 5 |- y e. _V
2	1	inex1 4594	. . . 4 |- ( y i^i A ) e. _V
3		ineq1 3698	. . . . 5 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( ( y i^i A ) i^i B ) )
4		inass 3713	. . . . 5 |- ( ( y i^i A ) i^i B ) = ( y i^i ( A i^i B ) )
5	3, 4	syl6eq 2524	. . . 4 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( y i^i ( A i^i B ) ) )
6	2, 5	abrexco 6155	. . 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 2467	. . . . . 6 |- ( y e. J |-> ( y i^i A ) ) = ( y e. J |-> ( y i^i A ) )
8	7	rnmpt 5254	. . . . 5 |- ran ( y e. J |-> ( y i^i A ) ) = { w | E. y e. J w = ( y i^i A ) }
9		mpteq1 4533	. . . . 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 5254	. . 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 2467	. . . 4 |- ( y e. J |-> ( y i^i ( A i^i B ) ) ) = ( y e. J |-> ( y i^i ( A i^i B ) ) )
13	12	rnmpt 5254	. . 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 2506	. 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 14699	. . . . 5 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
16	15	3adant3 1016	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
17	16	oveq1d 6310	. . 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 6320	. . . . 5 |- ( J ↾t A ) e. _V
19	16, 18	syl6eqelr 2564	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ran ( y e. J |-> ( y i^i A ) ) e. _V )
20		simp3 998	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> B e. X )
21		restval 14699	. . . 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 661	. . 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 2508	. 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 996	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> J e. V )
25		inex1g 4596	. . . 4 |- ( A e. W -> ( A i^i B ) e. _V )
26	25	3ad2ant2 1018	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( A i^i B ) e. _V )
27		restval 14699	. . 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 661	. 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 2534	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 973   = wceq 1379   e. wcel 1767   {cab 2452   E.wrex 2818   _Vcvv 3118   i^i cin 3480   |-> cmpt 4511   ran crn 5006  (class class class)co 6295   ↾_t crest 14693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-rest 14695

This theorem is referenced by:  restabs  19534  restin  19535  resstopn  19555  ressuss  20634