Theorem ssnei2 7730

Description: Any subset of X containing a neigborhood of a set is a neighborhood of this set. Based on Bourbaki TG I.3 Vi. (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 (_ M /\ M (_ X)) -> M e. ((nei` J)` S))

Proof of Theorem ssnei2

Step	Hyp	Ref	Expression
1		simprr 415	. . 3 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N (_ M /\ M (_ X)) -> M (_ X)
2		sstr2 2071	. . . . . . . . 9 |- (g (_ N -> (N (_ M -> g (_ M))
3	2	com12 11	. . . . . . . 8 |- (N (_ M -> (g (_ N -> g (_ M))
4	3	anim2d 561	. . . . . . 7 |- (N (_ M -> ((S (_ g /\ g (_ N) -> (S (_ g /\ g (_ M)))
5	4	r19.22sdv 1738	. . . . . 6 |- (N (_ M -> (E.g e. J (S (_ g /\ g (_ N) -> E.g e. J (S (_ g /\ g (_ M)))
6	5	impcom 351	. . . . 5 |- ((E.g e. J (S (_ g /\ g (_ N) /\ N (_ M) -> E.g e. J (S (_ g /\ g (_ M))
7		neii2 7722	. . . . 5 |- ((J e. Top /\ N e. ((nei` J)` S)) -> E.g e. J (S (_ g /\ g (_ N))
8	6, 7	sylan 448	. . . 4 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ N (_ M) -> E.g e. J (S (_ g /\ g (_ M))
9	8	adantrr 395	. . 3 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N (_ M /\ M (_ X)) -> E.g e. J (S (_ g /\ g (_ M))
10	1, 9	jca 288	. 2 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N (_ M /\ M (_ X)) -> (M (_ X /\ E.g e. J (S (_ g /\ g (_ M)))
11		neips.1	. . . . 5 |- X = U.J
12	11	neiss2 7716	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S (_ X)
13	11	isnei 7718	. . . 4 |- ((J e. Top /\ S (_ X) -> (M e. ((nei` J)` S) <-> (M (_ X /\ E.g e. J (S (_ g /\ g (_ M))))
14	12, 13	syldan 467	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> (M e. ((nei` J)` S) <-> (M (_ X /\ E.g e. J (S (_ g /\ g (_ M))))
15	14	adantr 389	. 2 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N (_ M /\ M (_ X)) -> (M e. ((nei` J)` S) <-> (M (_ X /\ E.g e. J (S (_ g /\ g (_ M))))
16	10, 15	mpbird 196	1 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (N (_ M /\ M (_ X)) -> M e. ((nei` J)` S))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wrex 1646   (_ wss 2047  U.cuni 2503  ` cfv 3182  Topctop 7588  neicnei 7712

This theorem is referenced by:  neifil 10568

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-nei 7713

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