Theorem restco 18727

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 2973	. . . . 5 |- y e. _V
2	1	inex1 4430	. . . 4 |- ( y i^i A ) e. _V
3		ineq1 3542	. . . . 5 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( ( y i^i A ) i^i B ) )
4		inass 3557	. . . . 5 |- ( ( y i^i A ) i^i B ) = ( y i^i ( A i^i B ) )
5	3, 4	syl6eq 2489	. . . 4 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( y i^i ( A i^i B ) ) )
6	2, 5	abrexco 5958	. . 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 2441	. . . . . 6 |- ( y e. J |-> ( y i^i A ) ) = ( y e. J |-> ( y i^i A ) )
8	7	rnmpt 5081	. . . . 5 |- ran ( y e. J |-> ( y i^i A ) ) = { w | E. y e. J w = ( y i^i A ) }
9		mpteq1 4369	. . . . 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 5081	. . 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 2441	. . . 4 |- ( y e. J |-> ( y i^i ( A i^i B ) ) ) = ( y e. J |-> ( y i^i ( A i^i B ) ) )
13	12	rnmpt 5081	. . 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 2471	. 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 14361	. . . . 5 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
16	15	3adant3 1003	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
17	16	oveq1d 6105	. . 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 6115	. . . . 5 |- ( J ↾t A ) e. _V
19	16, 18	syl6eqelr 2530	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ran ( y e. J |-> ( y i^i A ) ) e. _V )
20		simp3 985	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> B e. X )
21		restval 14361	. . . 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 656	. . 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 2473	. 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 983	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> J e. V )
25		inex1g 4432	. . . 4 |- ( A e. W -> ( A i^i B ) e. _V )
26	25	3ad2ant2 1005	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( A i^i B ) e. _V )
27		restval 14361	. . 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 656	. 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 2499	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 960   = wceq 1364   e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970   i^i cin 3324   e. cmpt 4347   ran crn 4837  (class class class)co 6090   ↾_t crest 14355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-rest 14357

This theorem is referenced by:  restabs  18728  restin  18729  resstopn  18749  ressuss  19797