Theorem restco 14761

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 2779	. . . . 5 |- y e. _V
2	1	inex1 4194	. . . 4 |- ( y i^i A ) e. _V
3		ineq1 3375	. . . . 5 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( ( y i^i A ) i^i B ) )
4		inass 3391	. . . . 5 |- ( ( y i^i A ) i^i B ) = ( y i^i ( A i^i B ) )
5	3, 4	eqtrdi 2256	. . . 4 |- ( x = ( y i^i A ) -> ( x i^i B ) = ( y i^i ( A i^i B ) ) )
6	2, 5	abrexco 5851	. . 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 2207	. . . . . 6 |- ( y e. J |-> ( y i^i A ) ) = ( y e. J |-> ( y i^i A ) )
8	7	rnmpt 4945	. . . . 5 |- ran ( y e. J |-> ( y i^i A ) ) = { w | E. y e. J w = ( y i^i A ) }
9		mpteq1 4144	. . . . 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 4945	. . 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 2207	. . . 4 |- ( y e. J |-> ( y i^i ( A i^i B ) ) ) = ( y e. J |-> ( y i^i ( A i^i B ) ) )
13	12	rnmpt 4945	. . 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 2238	. 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 13192	. . . . 5 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
16	15	3adant3 1020	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t A ) = ran ( y e. J |-> ( y i^i A ) ) )
17	16	oveq1d 5982	. . 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 13190	. . . . . 6 |- ↾t Fn ( _V X. _V )
19		simp1 1000	. . . . . . 7 |- ( ( J e. V /\ A e. W /\ B e. X ) -> J e. V )
20	19	elexd 2790	. . . . . 6 |- ( ( J e. V /\ A e. W /\ B e. X ) -> J e. _V )
21		simp2 1001	. . . . . . 7 |- ( ( J e. V /\ A e. W /\ B e. X ) -> A e. W )
22	21	elexd 2790	. . . . . 6 |- ( ( J e. V /\ A e. W /\ B e. X ) -> A e. _V )
23		fnovex 6000	. . . . . 6 |- ( ( ↾t Fn ( _V X. _V ) /\ J e. _V /\ A e. _V ) -> ( J ↾t A ) e. _V )
24	18, 20, 22, 23	mp3an2i 1355	. . . . 5 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( J ↾t A ) e. _V )
25	16, 24	eqeltrrd 2285	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ran ( y e. J |-> ( y i^i A ) ) e. _V )
26		simp3 1002	. . . 4 |- ( ( J e. V /\ A e. W /\ B e. X ) -> B e. X )
27		restval 13192	. . . 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 411	. . 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 2240	. 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 4196	. . . 4 |- ( A e. W -> ( A i^i B ) e. _V )
31	30	3ad2ant2 1022	. . 3 |- ( ( J e. V /\ A e. W /\ B e. X ) -> ( A i^i B ) e. _V )
32		restval 13192	. . 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 411	. 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 2266	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 981   = wceq 1373   e. wcel 2178   {cab 2193   E.wrex 2487   _Vcvv 2776   i^i cin 3173   |-> cmpt 4121   X. cxp 4691   ran crn 4694   Fn wfn 5285  (class class class)co 5967   ↾_t crest 13186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-rest 13188

This theorem is referenced by:  restabs  14762  restin  14763