Theorem ssrest 20980

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> x e. ( J ↾t A ) )
2		ssrexv 3667	. . . . . 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 763	. . . . 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		n0i 3920	. . . . . . . 8 |- ( x e. ( J ↾t A ) -> -. ( J ↾t A ) = (/) )
5		restfn 16085	. . . . . . . . . 10 |- ↾t Fn ( _V X. _V )
6		fndm 5990	. . . . . . . . . 10 |- ( ↾t Fn ( _V X. _V ) -> dom ↾t = ( _V X. _V ) )
7	5, 6	ax-mp 5	. . . . . . . . 9 |- dom ↾t = ( _V X. _V )
8	7	ndmov 6818	. . . . . . . 8 |- ( -. ( J e. _V /\ A e. _V ) -> ( J ↾t A ) = (/) )
9	4, 8	nsyl2 142	. . . . . . 7 |- ( x e. ( J ↾t A ) -> ( J e. _V /\ A e. _V ) )
10	9	adantl 482	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( J e. _V /\ A e. _V ) )
11		elrest 16088	. . . . . 6 |- ( ( J e. _V /\ A e. _V ) -> ( x e. ( J ↾t A ) <-> E. y e. J x = ( y i^i A ) ) )
12	10, 11	syl 17	. . . . 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 ) ) )
13		simpll 790	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> K e. V )
14	10	simprd 479	. . . . . 6 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> A e. _V )
15		elrest 16088	. . . . . 6 |- ( ( K e. V /\ A e. _V ) -> ( x e. ( K ↾t A ) <-> E. y e. K x = ( y i^i A ) ) )
16	13, 14, 15	syl2anc 693	. . . . 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 ) ) )
17	3, 12, 16	3imtr4d 283	. . . 4 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> ( x e. ( J ↾t A ) -> x e. ( K ↾t A ) ) )
18	1, 17	mpd 15	. . 3 |- ( ( ( K e. V /\ J C_ K ) /\ x e. ( J ↾t A ) ) -> x e. ( K ↾t A ) )
19	18	ex 450	. 2 |- ( ( K e. V /\ J C_ K ) -> ( x e. ( J ↾t A ) -> x e. ( K ↾t A ) ) )
20	19	ssrdv 3609	1 |- ( ( K e. V /\ J C_ K ) -> ( J ↾t A ) C_ ( K ↾t A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   X. cxp 5112   dom cdm 5114   Fn wfn 5883  (class class class)co 6650   ↾_t crest 16081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083

This theorem is referenced by:  1stcrest  21256  kgencmp  21348  kgencmp2  21349  kgen2ss  21358  ssufl  21722  cnfsmf  40949  smfsssmf  40952