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Theorem restid2 12139
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 4104 . . . . 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 4068 . . 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 12136 . . 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 3097 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  e.  ~P A )
98elpwid 3521 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  C_  A )
10 df-ss 3084 . . . . . . 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 4018 . . . . 5  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  ( x  e.  J  |->  x ) )
13 mptresid 4873 . . . . 5  |-  ( x  e.  J  |->  x )  =  (  _I  |`  J )
1412, 13syl6eq 2188 . . . 4  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  (  _I  |`  J ) )
1514rneqd 4768 . . 3  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  ran  (  _I  |`  J ) )
16 rnresi 4896 . . 3  |-  ran  (  _I  |`  J )  =  J
1715, 16syl6eq 2188 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  J )
187, 17eqtrd 2172 1  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   _Vcvv 2686    i^i cin 3070    C_ wss 3071   ~Pcpw 3510    |-> cmpt 3989    _I cid 4210   ran crn 4540    |` cres 4541  (class class class)co 5774   ↾t crest 12130
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-rest 12132
This theorem is referenced by:  restid  12141  topnidg  12143
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