Theorem ssrest 15047

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 ) -> ( J ↾t A ) C_ ( K ↾t A ) )

Proof of Theorem ssrest

Dummy variables x y j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> x e. ( J ↾t A ) )
2		ssrexv 3303	. . . . . 6 |- ( J C_ K -> ( E. y e. J x = ( y i^i A ) -> E. y e. K x = ( y i^i A ) ) )
3	2	ad2antlr 489	. . . . 5 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( E. y e. J x = ( y i^i A ) -> E. y e. K x = ( y i^i A ) ) )
4		df-rest 13454	. . . . . . . 8 |- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
5	4	elmpocl 6249	. . . . . . 7 |- ( x e. ( J ↾t A ) -> ( J e. _V /\ A e. _V ) )
6	5	adantl 277	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( J e. _V /\ A e. _V ) )
7		elrest 13459	. . . . . 6 |- ( ( J e. _V /\ A e. _V ) -> ( x e. ( J ↾t A ) <-> E. y e. J x = ( y i^i A ) ) )
8	6, 7	syl 14	. . . . 5 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( x e. ( J ↾t A ) <-> E. y e. J x = ( y i^i A ) ) )
9		simpll 527	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> K e. V )
10	6	simprd 114	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> A e. _V )
11		elrest 13459	. . . . . 6 |- ( ( K e. V /\ A e. _V ) -> ( x e. ( K ↾t A ) <-> E. y e. K x = ( y i^i A ) ) )
12	9, 10, 11	syl2anc 411	. . . . 5 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( x e. ( K ↾t A ) <-> E. y e. K x = ( y i^i A ) ) )
13	3, 8, 12	3imtr4d 203	. . . 4 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( x e. ( J ↾t A ) -> x e. ( K ↾t A ) ) )
14	1, 13	mpd 13	. . 3 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> x e. ( K ↾t A ) )
15	14	ex 115	. 2 |- ( ( K e. V /\ J C_ K ) -> ( x e. ( J ↾t A ) -> x e. ( K ↾t A ) ) )
16	15	ssrdv 3244	1 |- ( ( K e. V /\ J C_ K ) -> ( J ↾t A ) C_ ( K ↾t A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   E.wrex 2521   _Vcvv 2813   i^i cin 3210   C_ wss 3211   |-> cmpt 4171   ran crn 4750  (class class class)co 6050   ↾_t crest 13452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-rest 13454

This theorem is referenced by: (None)