Theorem ssrest 14361

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 3245	. . . . . 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 12855	. . . . . . . 8 |- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
5	4	elmpocl 6115	. . . . . . 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 12860	. . . . . 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 12860	. . . . . 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 3186	1 |- ( ( K e. V /\ J C_ K ) -> ( J ↾t A ) C_ ( K ↾t A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473   _Vcvv 2760   i^i cin 3153   C_ wss 3154   |-> cmpt 4091   ran crn 4661  (class class class)co 5919   ↾_t crest 12853

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-rest 12855

This theorem is referenced by: (None)