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Theorem restid2 11828
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 4036 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 271 . . . 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 4000 . . 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 11825 . . 3  |-  ( ( J  e.  _V  /\  A  e.  V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
74, 5, 6syl2anc 404 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
83sselda 3039 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  e.  ~P A )
98elpwid 3460 . . . . . . 7  |-  ( ( ( A  e.  V  /\  J  C_  ~P A
)  /\  x  e.  J )  ->  x  C_  A )
10 df-ss 3026 . . . . . . 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 3950 . . . . 5  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  ( x  e.  J  |->  x ) )
13 mptresid 4799 . . . . 5  |-  ( x  e.  J  |->  x )  =  (  _I  |`  J )
1412, 13syl6eq 2143 . . . 4  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( x  e.  J  |->  ( x  i^i 
A ) )  =  (  _I  |`  J ) )
1514rneqd 4696 . . 3  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  ran  (  _I  |`  J ) )
16 rnresi 4822 . . 3  |-  ran  (  _I  |`  J )  =  J
1715, 16syl6eq 2143 . 2  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ran  ( x  e.  J  |->  ( x  i^i  A ) )  =  J )
187, 17eqtrd 2127 1  |-  ( ( A  e.  V  /\  J  C_  ~P A )  ->  ( Jt  A )  =  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1296    e. wcel 1445   _Vcvv 2633    i^i cin 3012    C_ wss 3013   ~Pcpw 3449    |-> cmpt 3921    _I cid 4139   ran crn 4468    |` cres 4469  (class class class)co 5690   ↾t crest 11819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-rest 11821
This theorem is referenced by:  restid  11830  topnidg  11832
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