Theorem topssnei 12802

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 991	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> K e. Top )
2		simprl 521	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> J C_ K )
3		simpl1 990	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> J e. Top )
4		simprr 522	. . . . . . . 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 12787	. . . . . . . 8 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> x C_ X )
7	3, 4, 6	syl2anc 409	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> x C_ X )
8	5	ntropn 12757	. . . . . . 7 |- ( ( J e. Top /\ x C_ X ) -> ( ( int ` J ) ` x ) e. J )
9	3, 7, 8	syl2anc 409	. . . . . 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 3143	. . . . 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 12782	. . . . . . . 8 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> S C_ X )
12	3, 4, 11	syl2anc 409	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> S C_ X )
13	5	neiint 12785	. . . . . . 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 1228	. . . . . 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 146	. . . . 5 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> S C_ ( ( int ` J ) ` x ) )
16		opnneiss 12798	. . . . 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 1228	. . . 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 12761	. . . . 5 |- ( ( J e. Top /\ x C_ X ) -> ( ( int ` J ) ` x ) C_ x )
19	3, 7, 18	syl2anc 409	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> ( ( int ` J ) ` x ) C_ x )
20		simpl3 992	. . . . 5 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> X = Y )
21	7, 20	sseqtrd 3180	. . . 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 12797	. . . 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 1229	. . 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 373	. 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 3148	1 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ J C_ K ) -> ( ( nei ` J ) ` S ) C_ ( ( nei ` K ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   C_ wss 3116   U.cuni 3789   ` cfv 5188   Topctop 12635   intcnt 12733   neicnei 12778

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-ntr 12736  df-nei 12779

This theorem is referenced by: (None)