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Theorem restid2 12056
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 4074 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 274 . . . 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 )
42, 3ssexd 4038 . . 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 12053 . . 3  |-  ( ( J  e.  _V  /\  A  e.  V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
74, 5, 6syl2anc 408 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
83sselda 3067 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  e.  ~P A )
98elpwid 3491 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  C_  A )
10 df-ss 3054 . . . . . . 7  |-  ( x 
C_  A  <->  ( x  i^i  A )  =  x )
119, 10sylib 121 . . . . . 6  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  (
x  i^i  A )  =  x )
1211mpteq2dva 3988 . . . . 5  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  ( x  e.  J  |->  x ) )
13 mptresid 4843 . . . . 5  |-  ( x  e.  J  |->  x )  =  (  _I  |`  J )
1412, 13syl6eq 2166 . . . 4  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  (  _I  |`  J ) )
1514rneqd 4738 . . 3  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  ran  (  _I  |`  J ) )
16 rnresi 4866 . . 3  |-  ran  (  _I  |`  J )  =  J
1715, 16syl6eq 2166 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  J )
187, 17eqtrd 2150 1  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   _Vcvv 2660    i^i cin 3040    C_ wss 3041   ~Pcpw 3480    |-> cmpt 3959    _I cid 4180   ran crn 4510    |` cres 4511  (class class class)co 5742   ↾t crest 12047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-rest 12049
This theorem is referenced by:  restid  12058  topnidg  12060
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