Theorem topssnei 14634

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 1004	. . . 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 1003	. . . . . . 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 14619	. . . . . . . 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 14589	. . . . . . 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 3194	. . . . 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 14614	. . . . . . . 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 14617	. . . . . . 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 1250	. . . . . 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 14630	. . . . 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 1250	. . . 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 14593	. . . . 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 1005	. . . . 5 |- ( ( ( J e. Top /\ K e. Top /\ X = Y ) /\ ( J C_ K /\ x e. ( ( nei ` J ) ` S ) ) ) -> X = Y )
21	7, 20	sseqtrd 3231	. . . 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 14629	. . . 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 1251	. . 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 3199	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 981   = wceq 1373   e. wcel 2176   C_ wss 3166   U.cuni 3850   ` cfv 5271   Topctop 14469   intcnt 14565   neicnei 14610

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  df-nei 14611

This theorem is referenced by: (None)