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Theorem restval 13331
Description: The subspace topology induced by the topology  J on the set  A. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restval  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
Distinct variable groups:    x, A    x, J
Allowed substitution hints:    V( x)    W( x)

Proof of Theorem restval
Dummy variables  j 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2796 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 mptexg 5745 . . . . 5  |-  ( J  e.  _V  ->  (
x  e.  J  |->  ( x  i^i  A ) )  e.  _V )
4 rnexg 4940 . . . . 5  |-  ( ( x  e.  J  |->  ( x  i^i  A ) )  e.  _V  ->  ran  ( x  e.  J  |->  ( x  i^i  A
) )  e.  _V )
53, 4syl 15 . . . 4  |-  ( J  e.  _V  ->  ran  ( x  e.  J  |->  ( x  i^i  A
) )  e.  _V )
65adantr 451 . . 3  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ran  ( x  e.  J  |->  ( x  i^i 
A ) )  e. 
_V )
7 simpl 443 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  j  =  J )
8 simpr 447 . . . . . . 7  |-  ( ( j  =  J  /\  y  =  A )  ->  y  =  A )
98ineq2d 3370 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( x  i^i  A ) )
107, 9mpteq12dv 4098 . . . . 5  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  e.  j 
|->  ( x  i^i  y
) )  =  ( x  e.  J  |->  ( x  i^i  A ) ) )
1110rneqd 4906 . . . 4  |-  ( ( j  =  J  /\  y  =  A )  ->  ran  ( x  e.  j  |->  ( x  i^i  y ) )  =  ran  ( x  e.  J  |->  ( x  i^i 
A ) ) )
12 df-rest 13327 . . . 4  |-t  =  ( j  e.  _V ,  y  e. 
_V  |->  ran  ( x  e.  j  |->  ( x  i^i  y ) ) )
1311, 12ovmpt2ga 5977 . . 3  |-  ( ( J  e.  _V  /\  A  e.  _V  /\  ran  ( x  e.  J  |->  ( x  i^i  A
) )  e.  _V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A ) ) )
146, 13mpd3an3 1278 . 2  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
151, 2, 14syl2an 463 1  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    e. cmpt 4077   ran crn 4690  (class class class)co 5858   ↾t crest 13325
This theorem is referenced by:  elrest  13332  0rest  13334  restid2  13335  tgrest  16890  resttopon  16892  restco  16895  rest0  16900  restfpw  16910  ordtrest2  16934  1stcrest  17179  2ndcrest  17180  kgencmp  17240  xkoptsub  17348  trfilss  17584  trfg  17586  uzrest  17592  ellimc2  19227  limcflf  19231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-rest 13327
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