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Theorem restval 15318
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 3053 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 3053 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 mptexg 6133 . . . . 5  |-  ( J  e.  _V  ->  (
x  e.  J  |->  ( x  i^i  A ) )  e.  _V )
4 rnexg 6722 . . . . 5  |-  ( ( x  e.  J  |->  ( x  i^i  A ) )  e.  _V  ->  ran  ( x  e.  J  |->  ( x  i^i  A
) )  e.  _V )
53, 4syl 17 . . . 4  |-  ( J  e.  _V  ->  ran  ( x  e.  J  |->  ( x  i^i  A
) )  e.  _V )
65adantr 467 . . 3  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ran  ( x  e.  J  |->  ( x  i^i 
A ) )  e. 
_V )
7 simpl 459 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  j  =  J )
8 simpr 463 . . . . . . 7  |-  ( ( j  =  J  /\  y  =  A )  ->  y  =  A )
98ineq2d 3633 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( x  i^i  A ) )
107, 9mpteq12dv 4480 . . . . 5  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  e.  j 
|->  ( x  i^i  y
) )  =  ( x  e.  J  |->  ( x  i^i  A ) ) )
1110rneqd 5061 . . . 4  |-  ( ( j  =  J  /\  y  =  A )  ->  ran  ( x  e.  j  |->  ( x  i^i  y ) )  =  ran  ( x  e.  J  |->  ( x  i^i 
A ) ) )
12 df-rest 15314 . . . 4  |-t  =  ( j  e.  _V ,  y  e. 
_V  |->  ran  ( x  e.  j  |->  ( x  i^i  y ) ) )
1311, 12ovmpt2ga 6423 . . 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 1364 . 2  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
151, 2, 14syl2an 480 1  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886   _Vcvv 3044    i^i cin 3402    |-> cmpt 4460   ran crn 4834  (class class class)co 6288   ↾t crest 15312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-rest 15314
This theorem is referenced by:  elrest  15319  0rest  15321  restid2  15322  tgrest  20168  resttopon  20170  restco  20173  rest0  20178  restfpw  20188  neitr  20189  ordtrest2  20213  1stcrest  20461  2ndcrest  20462  kgencmp  20553  xkoptsub  20662  trfilss  20897  trfg  20899  uzrest  20905  restmetu  21578  ellimc2  22825  limcflf  22829  ordtrest2NEW  28722  ptrest  31932
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