Theorem restid2 12565

Description: The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
restid2	|- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )

Proof of Theorem restid2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 4159	. . . . 5 |- ( A e. V -> ~P A e. _V )
2	1	adantr 274	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> ~P A e. _V )
3		simpr 109	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> J C_ ~P A )
4	2, 3	ssexd 4122	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> J e. _V )
5		simpl 108	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> A e. V )
6		restval 12562	. . 3 |- ( ( J e. _V /\ A e. V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
7	4, 5, 6	syl2anc 409	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
8	3	sselda 3142	. . . . . . . 8 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x e. ~P A )
9	8	elpwid 3570	. . . . . . 7 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x C_ A )
10		df-ss 3129	. . . . . . 7 |- ( x C_ A <-> ( x i^i A ) = x )
11	9, 10	sylib 121	. . . . . 6 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> ( x i^i A ) = x )
12	11	mpteq2dva 4072	. . . . 5 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( x e. J |-> x ) )
13		mptresid 4938	. . . . 5 |- ( x e. J |-> x ) = ( _I |` J )
14	12, 13	eqtrdi 2215	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( _I |` J ) )
15	14	rneqd 4833	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = ran ( _I |` J ) )
16		rnresi 4961	. . 3 |- ran ( _I |` J ) = J
17	15, 16	eqtrdi 2215	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = J )
18	7, 17	eqtrd 2198	1 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116   ~Pcpw 3559   |-> cmpt 4043   _I cid 4266   ran crn 4605   |` cres 4606  (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:  restid  12567  topnidg  12569