Theorem restval 11908

Description: The subspace topology induced by the topology J on the set A. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)

Assertion

Ref	Expression
restval	|- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )

Distinct variable groups:   x, A   x, J

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem restval

Dummy variables j y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2652	. 2 |- ( J e. V -> J e. _V )
2		elex 2652	. 2 |- ( A e. W -> A e. _V )
3		mptexg 5577	. . . . 5 |- ( J e. _V -> ( x e. J |-> ( x i^i A ) ) e. _V )
4		rnexg 4740	. . . . 5 |- ( ( x e. J |-> ( x i^i A ) ) e. _V -> ran ( x e. J |-> ( x i^i A ) ) e. _V )
5	3, 4	syl 14	. . . 4 |- ( J e. _V -> ran ( x e. J |-> ( x i^i A ) ) e. _V )
6	5	adantr 272	. . 3 |- ( ( J e. _V /\ A e. _V ) -> ran ( x e. J |-> ( x i^i A ) ) e. _V )
7		simpl 108	. . . . . 6 |- ( ( j = J /\ y = A ) -> j = J )
8		simpr 109	. . . . . . 7 |- ( ( j = J /\ y = A ) -> y = A )
9	8	ineq2d 3224	. . . . . 6 |- ( ( j = J /\ y = A ) -> ( x i^i y ) = ( x i^i A ) )
10	7, 9	mpteq12dv 3950	. . . . 5 |- ( ( j = J /\ y = A ) -> ( x e. j |-> ( x i^i y ) ) = ( x e. J |-> ( x i^i A ) ) )
11	10	rneqd 4706	. . . 4 |- ( ( j = J /\ y = A ) -> ran ( x e. j |-> ( x i^i y ) ) = ran ( x e. J |-> ( x i^i A ) ) )
12		df-rest 11904	. . . 4 |- ↾t = ( j e. _V , y e. _V |-> ran ( x e. j |-> ( x i^i y ) ) )
13	11, 12	ovmpoga 5832	. . 3 |- ( ( J e. _V /\ A e. _V /\ ran ( x e. J |-> ( x i^i A ) ) e. _V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
14	6, 13	mpd3an3 1284	. 2 |- ( ( J e. _V /\ A e. _V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
15	1, 2, 14	syl2an 285	1 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   _Vcvv 2641   i^i cin 3020   |-> cmpt 3929   ran crn 4478  (class class class)co 5706   ↾_t crest 11902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-rest 11904

This theorem is referenced by:  elrest  11909  restid2  11911  tgrest  12120  resttopon  12122  restco  12125  rest0  12130