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Theorem ssrest 17232
Description: If  K is a finer topology than  J, then the subspace topologies induced by  A maintain this relationship. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
ssrest  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )

Proof of Theorem ssrest
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 448 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Jt  A
) )
2 ssrexv 3400 . . . . . 6  |-  ( J 
C_  K  ->  ( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
32ad2antlr 708 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( E. y  e.  J  x  =  ( y  i^i  A )  ->  E. y  e.  K  x  =  ( y  i^i  A ) ) )
4 n0i 3625 . . . . . . . 8  |-  ( x  e.  ( Jt  A )  ->  -.  ( Jt  A
)  =  (/) )
5 restfn 13644 . . . . . . . . . 10  |-t  Fn  ( _V  X.  _V )
6 fndm 5536 . . . . . . . . . 10  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
75, 6ax-mp 8 . . . . . . . . 9  |-  domt  =  ( _V  X.  _V )
87ndmov 6223 . . . . . . . 8  |-  ( -.  ( J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  =  (/) )
94, 8nsyl2 121 . . . . . . 7  |-  ( x  e.  ( Jt  A )  ->  ( J  e. 
_V  /\  A  e.  _V ) )
109adantl 453 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( J  e.  _V  /\  A  e.  _V )
)
11 elrest 13647 . . . . . 6  |-  ( ( J  e.  _V  /\  A  e.  _V )  ->  ( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
1210, 11syl 16 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  <->  E. y  e.  J  x  =  ( y  i^i  A
) ) )
13 simpll 731 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  K  e.  V )
1410simprd 450 . . . . . 6  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  A  e.  _V )
15 elrest 13647 . . . . . 6  |-  ( ( K  e.  V  /\  A  e.  _V )  ->  ( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
1613, 14, 15syl2anc 643 . . . . 5  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Kt  A )  <->  E. y  e.  K  x  =  ( y  i^i  A
) ) )
173, 12, 163imtr4d 260 . . . 4  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
181, 17mpd 15 . . 3  |-  ( ( ( K  e.  V  /\  J  C_  K )  /\  x  e.  ( Jt  A ) )  ->  x  e.  ( Kt  A
) )
1918ex 424 . 2  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( x  e.  ( Jt  A )  ->  x  e.  ( Kt  A ) ) )
2019ssrdv 3346 1  |-  ( ( K  e.  V  /\  J  C_  K )  -> 
( Jt  A )  C_  ( Kt  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620    X. cxp 4868   dom cdm 4870    Fn wfn 5441  (class class class)co 6073   ↾t crest 13640
This theorem is referenced by:  1stcrest  17508  kgencmp  17569  kgencmp2  17570  kgen2ss  17579  ssufl  17942
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-pow 4369  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-1st 6341  df-2nd 6342  df-rest 13642
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