Theorem ssnei2 14325

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 14317	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
3		sstr2 3186	. . . . . . 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 2595	. . . 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 14310	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )
11	9	isnei 14312	. . . 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 946	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 1364   e. wcel 2164   E.wrex 2473   C_ wss 3153   U.cuni 3835   ` cfv 5254   Topctop 14165   neicnei 14306

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-top 14166  df-nei 14307

This theorem is referenced by:  topssnei  14330