Theorem topssnei 14398

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 14383	. . . . . . . 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 14353	. . . . . . 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 3184	. . . . 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 14378	. . . . . . . 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 14381	. . . . . . 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 14394	. . . . 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 14357	. . . . 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 3221	. . . 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 14393	. . . 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 3189	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 2167   C_ wss 3157   U.cuni 3839   ` cfv 5258   Topctop 14233   intcnt 14329   neicnei 14374

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-top 14234  df-ntr 14332  df-nei 14375

This theorem is referenced by: (None)