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Theorem restval 13646
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 2956 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 2956 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 mptexg 5957 . . . . 5  |-  ( J  e.  _V  ->  (
x  e.  J  |->  ( x  i^i  A ) )  e.  _V )
4 rnexg 5123 . . . . 5  |-  ( ( x  e.  J  |->  ( x  i^i  A ) )  e.  _V  ->  ran  ( x  e.  J  |->  ( x  i^i  A
) )  e.  _V )
53, 4syl 16 . . . 4  |-  ( J  e.  _V  ->  ran  ( x  e.  J  |->  ( x  i^i  A
) )  e.  _V )
65adantr 452 . . 3  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ran  ( x  e.  J  |->  ( x  i^i 
A ) )  e. 
_V )
7 simpl 444 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  j  =  J )
8 simpr 448 . . . . . . 7  |-  ( ( j  =  J  /\  y  =  A )  ->  y  =  A )
98ineq2d 3534 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( x  i^i  A ) )
107, 9mpteq12dv 4279 . . . . 5  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  e.  j 
|->  ( x  i^i  y
) )  =  ( x  e.  J  |->  ( x  i^i  A ) ) )
1110rneqd 5089 . . . 4  |-  ( ( j  =  J  /\  y  =  A )  ->  ran  ( x  e.  j  |->  ( x  i^i  y ) )  =  ran  ( x  e.  J  |->  ( x  i^i 
A ) ) )
12 df-rest 13642 . . . 4  |-t  =  ( j  e.  _V ,  y  e. 
_V  |->  ran  ( x  e.  j  |->  ( x  i^i  y ) ) )
1311, 12ovmpt2ga 6195 . . 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 1280 . 2  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
151, 2, 14syl2an 464 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311    e. cmpt 4258   ran crn 4871  (class class class)co 6073   ↾t crest 13640
This theorem is referenced by:  elrest  13647  0rest  13649  restid2  13650  tgrest  17215  resttopon  17217  restco  17220  rest0  17225  restfpw  17235  neitr  17236  ordtrest2  17260  1stcrest  17508  2ndcrest  17509  kgencmp  17569  xkoptsub  17678  trfilss  17913  trfg  17915  uzrest  17921  restmetu  18609  ellimc2  19756  limcflf  19760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-rest 13642
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