Theorem ssnei2 13519

Description: Any subset M of X containing a neighborhood N of a set S is a neighborhood of this set. Generalization to subsets of Property V_i of [BourbakiTop1] p. I.3. (Contributed by FL, 2-Oct-2006.)

Hypothesis

Ref	Expression
neips.1	|- X = U. J

Assertion

Ref	Expression
ssnei2	|- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> M e. ( ( nei ` J ) ` S ) )

Proof of Theorem ssnei2

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 531	. 2 |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> M C_ X )
2		neii2 13511	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
3		sstr2 3162	. . . . . . 7 |- ( g C_ N -> ( N C_ M -> g C_ M ) )
4	3	com12 30	. . . . . 6 |- ( N C_ M -> ( g C_ N -> g C_ M ) )
5	4	anim2d 337	. . . . 5 |- ( N C_ M -> ( ( S C_ g /\ g C_ N ) -> ( S C_ g /\ g C_ M ) ) )
6	5	reximdv 2578	. . . 4 |- ( N C_ M -> ( E. g e. J ( S C_ g /\ g C_ N ) -> E. g e. J ( S C_ g /\ g C_ M ) ) )
7	2, 6	mpan9 281	. . 3 |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ N C_ M ) -> E. g e. J ( S C_ g /\ g C_ M ) )
8	7	adantrr 479	. 2 |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> E. g e. J ( S C_ g /\ g C_ M ) )
9		neips.1	. . . . 5 |- X = U. J
10	9	neiss2 13504	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )
11	9	isnei 13506	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( M e. ( ( nei ` J ) ` S ) <-> ( M C_ X /\ E. g e. J ( S C_ g /\ g C_ M ) ) ) )
12	10, 11	syldan 282	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> ( M e. ( ( nei ` J ) ` S ) <-> ( M C_ X /\ E. g e. J ( S C_ g /\ g C_ M ) ) ) )
13	12	adantr 276	. 2 |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> ( M e. ( ( nei ` J ) ` S ) <-> ( M C_ X /\ E. g e. J ( S C_ g /\ g C_ M ) ) ) )
14	1, 8, 13	mpbir2and 944	1 |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> M e. ( ( nei ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   C_ wss 3129   U.cuni 3809   ` cfv 5214   Topctop 13357   neicnei 13500

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-top 13358  df-nei 13501

This theorem is referenced by:  topssnei  13524