Theorem ssnei2 14714

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 14706	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ N ) )
3		sstr2 3204	. . . . . . 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 2608	. . . 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 14699	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )
11	9	isnei 14701	. . . 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 947	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 1373   e. wcel 2177   E.wrex 2486   C_ wss 3170   U.cuni 3859   ` cfv 5285   Topctop 14554   neicnei 14695

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-top 14555  df-nei 14696

This theorem is referenced by:  topssnei  14719