Theorem restid2 12683

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 4178	. . . . 5 |- ( A e. V -> ~P A e. _V )
2	1	adantr 276	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> ~P A e. _V )
3		simpr 110	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> J C_ ~P A )
4	2, 3	ssexd 4141	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> J e. _V )
5		simpl 109	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> A e. V )
6		restval 12680	. . 3 |- ( ( J e. _V /\ A e. V ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
7	4, 5, 6	syl2anc 411	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = ran ( x e. J |-> ( x i^i A ) ) )
8	3	sselda 3155	. . . . . . . 8 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x e. ~P A )
9	8	elpwid 3586	. . . . . . 7 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x C_ A )
10		df-ss 3142	. . . . . . 7 |- ( x C_ A <-> ( x i^i A ) = x )
11	9, 10	sylib 122	. . . . . 6 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> ( x i^i A ) = x )
12	11	mpteq2dva 4091	. . . . 5 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( x e. J |-> x ) )
13		mptresid 4958	. . . . 5 |- ( x e. J |-> x ) = ( _I |` J )
14	12, 13	eqtrdi 2226	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( _I |` J ) )
15	14	rneqd 4853	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = ran ( _I |` J ) )
16		rnresi 4982	. . 3 |- ran ( _I |` J ) = J
17	15, 16	eqtrdi 2226	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = J )
18	7, 17	eqtrd 2210	1 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2737   i^i cin 3128   C_ wss 3129   ~Pcpw 3575   |-> cmpt 4062   _I cid 4286   ran crn 4625   |` cres 4626  (class class class)co 5870   ↾_t crest 12674

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-ov 5873  df-oprab 5874  df-mpo 5875  df-rest 12676

This theorem is referenced by:  restid  12685  topnidg  12687