Theorem restval 12562

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 2737	. 2 |- ( J e. V -> J e. _V )
2		elex 2737	. 2 |- ( A e. W -> A e. _V )
3		mptexg 5710	. . . . 5 |- ( J e. _V -> ( x e. J |-> ( x i^i A ) ) e. _V )
4		rnexg 4869	. . . . 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 274	. . 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 3323	. . . . . 6 |- ( ( j = J /\ y = A ) -> ( x i^i y ) = ( x i^i A ) )
10	7, 9	mpteq12dv 4064	. . . . 5 |- ( ( j = J /\ y = A ) -> ( x e. j |-> ( x i^i y ) ) = ( x e. J |-> ( x i^i A ) ) )
11	10	rneqd 4833	. . . 4 |- ( ( j = J /\ y = A ) -> ran ( x e. j |-> ( x i^i y ) ) = ran ( x e. J |-> ( x i^i A ) ) )
12		df-rest 12558	. . . 4 |- ↾t = ( j e. _V , y e. _V |-> ran ( x e. j |-> ( x i^i y ) ) )
13	11, 12	ovmpoga 5971	. . 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 1328	. 2 |- ( ( J e. _V /\ A e. _V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
15	1, 2, 14	syl2an 287	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 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   |-> cmpt 4043   ran crn 4605  (class class class)co 5842   ↾_t crest 12556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-rest 12558

This theorem is referenced by:  elrest  12563  restid2  12565  tgrest  12809  resttopon  12811  restco  12814  rest0  12819