Theorem restval 13318

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 2812	. 2 |- ( J e. V -> J e. _V )
2		elex 2812	. 2 |- ( A e. W -> A e. _V )
3		mptexg 5874	. . . . 5 |- ( J e. _V -> ( x e. J |-> ( x i^i A ) ) e. _V )
4		rnexg 4995	. . . . 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 276	. . 3 |- ( ( J e. _V /\ A e. _V ) -> ran ( x e. J |-> ( x i^i A ) ) e. _V )
7		simpl 109	. . . . . 6 |- ( ( j = J /\ y = A ) -> j = J )
8		simpr 110	. . . . . . 7 |- ( ( j = J /\ y = A ) -> y = A )
9	8	ineq2d 3406	. . . . . 6 |- ( ( j = J /\ y = A ) -> ( x i^i y ) = ( x i^i A ) )
10	7, 9	mpteq12dv 4169	. . . . 5 |- ( ( j = J /\ y = A ) -> ( x e. j |-> ( x i^i y ) ) = ( x e. J |-> ( x i^i A ) ) )
11	10	rneqd 4959	. . . 4 |- ( ( j = J /\ y = A ) -> ran ( x e. j |-> ( x i^i y ) ) = ran ( x e. J |-> ( x i^i A ) ) )
12		df-rest 13314	. . . 4 |- ↾t = ( j e. _V , y e. _V |-> ran ( x e. j |-> ( x i^i y ) ) )
13	11, 12	ovmpoga 6146	. . 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 1372	. 2 |- ( ( J e. _V /\ A e. _V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
15	1, 2, 14	syl2an 289	1 |- ( ( J e. V /\ A e. W ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2800   i^i cin 3197   |-> cmpt 4148   ran crn 4724  (class class class)co 6013   ↾_t crest 13312

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-rest 13314

This theorem is referenced by:  elrest  13319  restid2  13321  tgrest  14883  resttopon  14885  restco  14888  rest0  14893