Theorem topssnei 15186

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 1032	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> K e. Top )
2		simprl 535	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> J C_ K )
3		simpl1 1031	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> J e. Top )
4		simprr 537	. . . . . . . 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 15171	. . . . . . . 8 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> x C_ X )
7	3, 4, 6	syl2anc 415	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> x C_ X )
8	5	ntropn 15141	. . . . . . 7 |- ( ( J e. Top /\ x C_ X ) -> ( ( int ` J ) ` x ) e. J )
9	3, 7, 8	syl2anc 415	. . . . . 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 3249	. . . . 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 15166	. . . . . . . 8 |- ( ( J e. Top /\ x e. ( ( nei ` J ) ` S ) ) -> S C_ X )
12	3, 4, 11	syl2anc 415	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> S C_ X )
13	5	neiint 15169	. . . . . . 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 1278	. . . . . 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 15182	. . . . 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 1278	. . . 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 15145	. . . . 5 |- ( ( J e. Top /\ x C_ X ) -> ( ( int ` J ) ` x ) C_ x )
19	3, 7, 18	syl2anc 415	. . . 4 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> ( ( int ` J ) ` x ) C_ x )
20		simpl3 1033	. . . . 5 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> X = Y )
21	7, 20	sseqtrd 3286	. . . 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 15181	. . . 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 1279	. . 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 3254	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 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   U.cuni 3930   ` cfv 5372   Topctop 15021   intcnt 15117   neicnei 15162

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-top 15022  df-ntr 15120  df-nei 15163

This theorem is referenced by: (None)