ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  restid2 Unicode version

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 )  ->  ( Jt  A )  =  J )

Proof of Theorem restid2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pwexg 4178 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 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 )
42, 3ssexd 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 )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
74, 5, 6syl2anc 411 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
83sselda 3155 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  e.  ~P A )
98elpwid 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 )
119, 10sylib 122 . . . . . 6  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  (
x  i^i  A )  =  x )
1211mpteq2dva 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 )
1412, 13eqtrdi 2226 . . . 4  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  (  _I  |`  J ) )
1514rneqd 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
1715, 16eqtrdi 2226 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  J )
187, 17eqtrd 2210 1  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
Colors of variables: wff set class
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
  Copyright terms: Public domain W3C validator