Theorem neiss2 7716

Description: A set with a neighborhood is a subset of the topology's base set. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.)

Hypothesis

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

Assertion

Ref	Expression
neiss2	|- ((J e. Top /\ N e. ((nei` J)` S)) -> S (_ X)

Proof of Theorem neiss2

Step	Hyp	Ref	Expression
1		elfvdm 3747	. . . 4 |- (N e. ((nei` J)` S) -> S e. dom (nei` J))
2	1	adantl 388	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S e. dom (nei` J))
3		neifval.1	. . . . . . 7 |- X = U.J
4	3	neif 7715	. . . . . 6 |- (J e. Top -> (nei` J) Fn P~X)
5		fndm 3587	. . . . . 6 |- ((nei` J) Fn P~X -> dom (nei` J) = P~X)
6	4, 5	syl 10	. . . . 5 |- (J e. Top -> dom (nei` J) = P~X)
7	6	eleq2d 1541	. . . 4 |- (J e. Top -> (S e. dom (nei` J) <-> S e. P~X))
8	7	adantr 389	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> (S e. dom (nei` J) <-> S e. P~X))
9	2, 8	mpbid 195	. 2 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S e. P~X)
10		uniexg 2871	. . . . 5 |- (J e. Top -> U.J e. V)
11	10, 3	syl5eqel 1552	. . . 4 |- (J e. Top -> X e. V)
12		elpw2g 2727	. . . 4 |- (X e. V -> (S e. P~X <-> S (_ X))
13	11, 12	syl 10	. . 3 |- (J e. Top -> (S e. P~X <-> S (_ X))
14	13	adantr 389	. 2 |- ((J e. Top /\ N e. ((nei` J)` S)) -> (S e. P~X <-> S (_ X))
15	9, 14	mpbid 195	1 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S (_ X)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   (_ wss 2047  P~cpw 2401  U.cuni 2503  dom cdm 3170   Fn wfn 3177  ` cfv 3182  Topctop 7588  neicnei 7712

This theorem is referenced by:  neii1 7721  neii2 7722  neiss 7723  ssnei2 7730  innei 7736

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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