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Theorem restval 14844
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 3118 . 2  |-  ( J  e.  V  ->  J  e.  _V )
2 elex 3118 . 2  |-  ( A  e.  W  ->  A  e.  _V )
3 mptexg 6143 . . . . 5  |-  ( J  e.  _V  ->  (
x  e.  J  |->  ( x  i^i  A ) )  e.  _V )
4 rnexg 6731 . . . . 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 465 . . 3  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ran  ( x  e.  J  |->  ( x  i^i 
A ) )  e. 
_V )
7 simpl 457 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  j  =  J )
8 simpr 461 . . . . . . 7  |-  ( ( j  =  J  /\  y  =  A )  ->  y  =  A )
98ineq2d 3696 . . . . . 6  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  i^i  y
)  =  ( x  i^i  A ) )
107, 9mpteq12dv 4535 . . . . 5  |-  ( ( j  =  J  /\  y  =  A )  ->  ( x  e.  j 
|->  ( x  i^i  y
) )  =  ( x  e.  J  |->  ( x  i^i  A ) ) )
1110rneqd 5240 . . . 4  |-  ( ( j  =  J  /\  y  =  A )  ->  ran  ( x  e.  j  |->  ( x  i^i  y ) )  =  ran  ( x  e.  J  |->  ( x  i^i 
A ) ) )
12 df-rest 14840 . . . 4  |-t  =  ( j  e.  _V ,  y  e. 
_V  |->  ran  ( x  e.  j  |->  ( x  i^i  y ) ) )
1311, 12ovmpt2ga 6431 . . 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 1325 . 2  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  ran  ( x  e.  J  |->  ( x  i^i  A
) ) )
151, 2, 14syl2an 477 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 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    |-> cmpt 4515   ran crn 5009  (class class class)co 6296   ↾t crest 14838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-rest 14840
This theorem is referenced by:  elrest  14845  0rest  14847  restid2  14848  tgrest  19787  resttopon  19789  restco  19792  rest0  19797  restfpw  19807  neitr  19808  ordtrest2  19832  1stcrest  20080  2ndcrest  20081  kgencmp  20172  xkoptsub  20281  trfilss  20516  trfg  20518  uzrest  20524  restmetu  21216  ellimc2  22407  limcflf  22411  ordtrest2NEW  28066  ptrest  30232
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