Theorem restid2 13195

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 4240	. . . . 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 4200	. . 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 13192	. . 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 3201	. . . . . . . 8 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x e. ~P A )
9	8	elpwid 3637	. . . . . . 7 |- ( ( ( A e. V /\ J C_ ~P A ) /\ x e. J ) -> x C_ A )
10		df-ss 3187	. . . . . . 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 4150	. . . . 5 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( x e. J |-> x ) )
13		mptresid 5032	. . . . 5 |- ( _I |` J ) = ( x e. J |-> x )
14	12, 13	eqtr4di 2258	. . . 4 |- ( ( A e. V /\ J C_ ~P A ) -> ( x e. J |-> ( x i^i A ) ) = ( _I |` J ) )
15	14	rneqd 4926	. . 3 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = ran ( _I |` J ) )
16		rnresi 5058	. . 3 |- ran ( _I |` J ) = J
17	15, 16	eqtrdi 2256	. 2 |- ( ( A e. V /\ J C_ ~P A ) -> ran ( x e. J |-> ( x i^i A ) ) = J )
18	7, 17	eqtrd 2240	1 |- ( ( A e. V /\ J C_ ~P A ) -> ( J ↾t A ) = J )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   C_ wss 3174   ~Pcpw 3626   |-> cmpt 4121   _I cid 4353   ran crn 4694   |` cres 4695  (class class class)co 5967   ↾_t crest 13186

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-rest 13188

This theorem is referenced by:  restid  13197  topnidg  13199