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Theorem restval 13347
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 2809 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2809 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 mptexg 5761 . . . . 5  |-  ( J  e.  _V  ->  (
x  e.  J  |->  ( x  i^i  A ) )  e.  _V )
4 rnexg 4956 . . . . 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 3383 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( x  i^i  A ) )
107, 9mpteq12dv 4114 . . . . 5  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  e.  j 
|->  ( x  i^i  y
) )  =  ( x  e.  J  |->  ( x  i^i  A ) ) )
1110rneqd 4922 . . . 4  |-  ( ( j  =  J  /\  y  =  A )  ->  ran  ( x  e.  j  |->  ( x  i^i  y ) )  =  ran  ( x  e.  J  |->  ( x  i^i 
A ) ) )
12 df-rest 13343 . . . 4  |-t  =  ( j  e.  _V ,  y  e. 
_V  |->  ran  ( x  e.  j  |->  ( x  i^i  y ) ) )
1311, 12ovmpt2ga 5993 . . 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 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164    e. cmpt 4093   ran crn 4706  (class class class)co 5874   ↾t crest 13341
This theorem is referenced by:  elrest  13348  0rest  13350  restid2  13351  tgrest  16906  resttopon  16908  restco  16911  rest0  16916  restfpw  16926  ordtrest2  16950  1stcrest  17195  2ndcrest  17196  kgencmp  17256  xkoptsub  17364  trfilss  17600  trfg  17602  uzrest  17608  ellimc2  19243  limcflf  19247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-rest 13343
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