Theorem restid2 11828

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 4036	. . . . 5 |- ( A e. V -> ~P A e. _V )
2	1	adantr 271	. . . 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 4000	. . 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 11825	. . 3 |- ( ( J e. _V /\ A e. V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
7	4, 5, 6	syl2anc 404	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
8	3	sselda 3039	. . . . . . . 8 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x e. ~P A )
9	8	elpwid 3460	. . . . . . 7 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x C_ A )
10		df-ss 3026	. . . . . . 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 3950	. . . . 5 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( x e. J |-> x ) )
13		mptresid 4799	. . . . 5 |- ( x e. J |-> x ) = ( _I |` J )
14	12, 13	syl6eq 2143	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( _I |` J ) )
15	14	rneqd 4696	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = ran ( _I |` J ) )
16		rnresi 4822	. . 3 |- ran ( _I |` J ) = J
17	15, 16	syl6eq 2143	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = J )
18	7, 17	eqtrd 2127	1 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1296   e. wcel 1445   _Vcvv 2633   i^i cin 3012   C_ wss 3013   ~Pcpw 3449   |-> cmpt 3921   _I cid 4139   ran crn 4468   |` cres 4469  (class class class)co 5690   ↾_t crest 11819

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 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-rest 11821

This theorem is referenced by:  restid  11830  topnidg  11832