Theorem ssrest 12354

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 109	. . . 4 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> x e. ( J ↾t A ) )
2		ssrexv 3162	. . . . . 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 480	. . . . 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 12125	. . . . . . . 8 |- ↾t = ( j e. _V , x e. _V |-> ran ( y e. j |-> ( y i^i x ) ) )
5	4	elmpocl 5968	. . . . . . 7 |- ( x e. ( J ↾t A ) -> ( J e. _V /\ A e. _V ) )
6	5	adantl 275	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( J e. _V /\ A e. _V ) )
7		elrest 12130	. . . . . 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 518	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> K e. V )
10	6	simprd 113	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> A e. _V )
11		elrest 12130	. . . . . 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 408	. . . . 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 202	. . . 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 114	. 2 |- ( ( K e. V /\ J C_ K ) -> ( x e. ( J ↾t A ) -> x e. ( K ↾t A ) ) )
16	15	ssrdv 3103	1 |- ( ( K e. V /\ J C_ K ) -> ( J ↾t A ) C_ ( K ↾t A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   _Vcvv 2686   i^i cin 3070   C_ wss 3071   |-> cmpt 3989   ran crn 4540  (class class class)co 5774   ↾_t crest 12123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-rest 12125

This theorem is referenced by: (None)