Theorem restid2 13394

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 4276	. . . . 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 4234	. . 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 13391	. . 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 3228	. . . . . . . 8 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x e. ~P A )
9	8	elpwid 3667	. . . . . . 7 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x C_ A )
10		df-ss 3214	. . . . . . 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 4184	. . . . 5 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( x e. J |-> x ) )
13		mptresid 5073	. . . . 5 |- ( _I |` J ) = ( x e. J |-> x )
14	12, 13	eqtr4di 2282	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( _I |` J ) )
15	14	rneqd 4967	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = ran ( _I |` J ) )
16		rnresi 5100	. . 3 |- ran ( _I |` J ) = J
17	15, 16	eqtrdi 2280	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = J )
18	7, 17	eqtrd 2264	1 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   C_ wss 3201   ~Pcpw 3656   |-> cmpt 4155   _I cid 4391   ran crn 4732   |` cres 4733  (class class class)co 6028   ↾_t crest 13385

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-rest 13387

This theorem is referenced by:  restid  13396  topnidg  13398