Theorem restval 12916

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 2774	. 2 |- ( J e. V -> J e. _V )
2		elex 2774	. 2 |- ( A e. W -> A e. _V )
3		mptexg 5787	. . . . 5 |- ( J e. _V -> ( x e. J |-> ( x i^i A ) ) e. _V )
4		rnexg 4931	. . . . 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 3364	. . . . . 6 |- ( ( j = J /\ y = A ) -> ( x i^i y ) = ( x i^i A ) )
10	7, 9	mpteq12dv 4115	. . . . 5 |- ( ( j = J /\ y = A ) -> ( x e. j |-> ( x i^i y ) ) = ( x e. J |-> ( x i^i A ) ) )
11	10	rneqd 4895	. . . 4 |- ( ( j = J /\ y = A ) -> ran ( x e. j |-> ( x i^i y ) ) = ran ( x e. J |-> ( x i^i A ) ) )
12		df-rest 12912	. . . 4 |- ↾t = ( j e. _V , y e. _V |-> ran ( x e. j |-> ( x i^i y ) ) )
13	11, 12	ovmpoga 6052	. . 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 1349	. 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 1364   e. wcel 2167   _Vcvv 2763   i^i cin 3156   |-> cmpt 4094   ran crn 4664  (class class class)co 5922   ↾_t crest 12910

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-rest 12912

This theorem is referenced by:  elrest  12917  restid2  12919  tgrest  14405  resttopon  14407  restco  14410  rest0  14415