Theorem ssntr 14594

Description: An open subset of a set is a subset of the set's interior. (Contributed by Jeff Hankins, 31-Aug-2009.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
clscld.1	|- X = U. J

Assertion

Ref	Expression
ssntr	|- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ ( ( int ` J ) ` S ) )

Proof of Theorem ssntr

Step	Hyp	Ref	Expression
1		elin 3356	. . . . 5 |- ( O e. ( J i^i ~P S ) <-> ( O e. J /\ O e. ~P S ) )
2		elpwg 3624	. . . . . 6 |- ( O e. J -> ( O e. ~P S <-> O C_ S ) )
3	2	pm5.32i 454	. . . . 5 |- ( ( O e. J /\ O e. ~P S ) <-> ( O e. J /\ O C_ S ) )
4	1, 3	bitr2i 185	. . . 4 |- ( ( O e. J /\ O C_ S ) <-> O e. ( J i^i ~P S ) )
5		elssuni 3878	. . . 4 |- ( O e. ( J i^i ~P S ) -> O C_ U. ( J i^i ~P S ) )
6	4, 5	sylbi 121	. . 3 |- ( ( O e. J /\ O C_ S ) -> O C_ U. ( J i^i ~P S ) )
7	6	adantl 277	. 2 |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ U. ( J i^i ~P S ) )
8		clscld.1	. . . 4 |- X = U. J
9	8	ntrval 14582	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
10	9	adantr 276	. 2 |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
11	7, 10	sseqtrrd 3232	1 |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ ( ( int ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   i^i cin 3165   C_ wss 3166   ~Pcpw 3616   U.cuni 3850   ` cfv 5271   Topctop 14469   intcnt 14565

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 14470  df-ntr 14568

This theorem is referenced by:  ntrin  14596  neiint  14617  cnntri  14696