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Theorem restid2 13650
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 )  ->  ( Jt  A )  =  J )

Proof of Theorem restid2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pwexg 4375 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 452 . . . 4  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ~P A  e. 
_V )
3 simpr 448 . . . 4  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  J  C_  ~P A )
42, 3ssexd 4342 . . 3  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  J  e.  _V )
5 simpl 444 . . 3  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  A  e.  V
)
6 restval 13646 . . 3  |-  ( ( J  e.  _V  /\  A  e.  V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
74, 5, 6syl2anc 643 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
83sselda 3340 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  e.  ~P A )
98elpwid 3800 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  C_  A )
10 df-ss 3326 . . . . . . 7  |-  ( x 
C_  A  <->  ( x  i^i  A )  =  x )
119, 10sylib 189 . . . . . 6  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  (
x  i^i  A )  =  x )
1211mpteq2dva 4287 . . . . 5  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  ( x  e.  J  |->  x ) )
13 mptresid 5187 . . . . 5  |-  ( x  e.  J  |->  x )  =  (  _I  |`  J )
1412, 13syl6eq 2483 . . . 4  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  (  _I  |`  J ) )
1514rneqd 5089 . . 3  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  ran  (  _I  |`  J ) )
16 rnresi 5211 . . 3  |-  ran  (  _I  |`  J )  =  J
1715, 16syl6eq 2483 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  J )
187, 17eqtrd 2467 1  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791    e. cmpt 4258    _I cid 4485   ran crn 4871    |` cres 4872  (class class class)co 6073   ↾t crest 13640
This theorem is referenced by:  restid  13653  topnid  13655  ssufl  17942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-rest 13642
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