Theorem ssnei2 14880

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 533	. 2 |- ( ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) /\ ( N C_ M /\ M C_ X ) ) -> M C_ X )
2		neii2 14872	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
3		sstr2 3234	. . . . . . 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 2633	. . . 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 14865	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )
11	9	isnei 14867	. . . 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 952	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 1397   e. wcel 2202   E.wrex 2511   C_ wss 3200   U.cuni 3893   ` cfv 5326   Topctop 14720   neicnei 14861

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-top 14721  df-nei 14862

This theorem is referenced by:  topssnei  14885