Theorem ssrest 14850

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 3289	. . . . . 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 13269	. . . . . . . 8 |- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
5	4	elmpocl 6199	. . . . . . 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 13274	. . . . . 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 13274	. . . . . 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 3230	1 |- ( ( K e. V /\ J C_ K ) -> ( J ↾t A ) C_ ( K ↾t A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   _Vcvv 2799   i^i cin 3196   C_ wss 3197   |-> cmpt 4144   ran crn 4719  (class class class)co 6000   ↾_t crest 13267

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-rest 13269

This theorem is referenced by: (None)