Theorem topssnei 14330

Description: A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)

Hypotheses

Ref	Expression
tpnei.1	|- X = U. J
topssnei.2	|- Y = U. K

Assertion

Ref	Expression
topssnei	|- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ J C_ K ) -> ( ( nei ` J ) ` S ) C_ ( ( nei ` K ) ` S ) )

Proof of Theorem topssnei

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1003	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> K e. Top )
2		simprl 529	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> J C_ K )
3		simpl1 1002	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> J e. Top )
4		simprr 531	. . . . . . . 8 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> x e. ( ( nei ` J ) ` S ) )
5		tpnei.1	. . . . . . . . 9 |- X = U. J
6	5	neii1 14315	. . . . . . . 8 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> x C_ X )
7	3, 4, 6	syl2anc 411	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> x C_ X )
8	5	ntropn 14285	. . . . . . 7 |- ( ( J e. Top /\ x C_ X ) -> ( ( int ` J ) ` x ) e. J )
9	3, 7, 8	syl2anc 411	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> ( ( int ` J ) ` x ) e. J )
10	2, 9	sseldd 3180	. . . . 5 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> ( ( int ` J ) ` x ) e. K )
11	5	neiss2 14310	. . . . . . . 8 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> S C_ X )
12	3, 4, 11	syl2anc 411	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> S C_ X )
13	5	neiint 14313	. . . . . . 7 |- ( ( J e. Top /\ S C_ X /\ x C_ X ) -> ( x e. ( ( nei ` J ) ` S ) <-> S C_ ( ( int ` J ) ` x ) ) )
14	3, 12, 7, 13	syl3anc 1249	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> ( x e. ( ( nei ` J ) ` S ) <-> S C_ ( ( int ` J ) ` x ) ) )
15	4, 14	mpbid 147	. . . . 5 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> S C_ ( ( int ` J ) ` x ) )
16		opnneiss 14326	. . . . 5 |- ( ( K e. Top /\ ( ( int ` J ) ` x ) e. K /\ S C_ ( ( int ` J ) ` x ) ) -> ( ( int ` J ) ` x ) e. ( ( nei ` K ) ` S ) )
17	1, 10, 15, 16	syl3anc 1249	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> ( ( int ` J ) ` x ) e. ( ( nei ` K ) ` S ) )
18	5	ntrss2 14289	. . . . 5 |- ( ( J e. Top /\ x C_ X ) -> ( ( int ` J ) ` x ) C_ x )
19	3, 7, 18	syl2anc 411	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> ( ( int ` J ) ` x ) C_ x )
20		simpl3 1004	. . . . 5 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> X = Y )
21	7, 20	sseqtrd 3217	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> x C_ Y )
22		topssnei.2	. . . . 5 |- Y = U. K
23	22	ssnei2 14325	. . . 4 |- ( ( ( K e. Top /\ ( ( int ` J ) ` x ) e. ( ( nei ` K ) ` S ) ) /\ ( ( ( int ` J ) ` x ) C_ x /\ x C_ Y ) ) -> x e. ( ( nei ` K ) ` S ) )
24	1, 17, 19, 21, 23	syl22anc 1250	. . 3 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> x e. ( ( nei ` K ) ` S ) )
25	24	expr 375	. 2 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ J C_ K ) -> ( x e. ( ( nei ` J ) ` S ) -> x e. ( ( nei ` K ) ` S ) ) )
26	25	ssrdv 3185	1 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ J C_ K ) -> ( ( nei ` J ) ` S ) C_ ( ( nei ` K ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   C_ wss 3153   U.cuni 3835   ` cfv 5254   Topctop 14165   intcnt 14261   neicnei 14306

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 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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  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-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-top 14166  df-ntr 14264  df-nei 14307

This theorem is referenced by: (None)