Theorem ntrval2 7686

Description: Interior expressed in terms of closure.

Hypothesis

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

Assertion

Ref	Expression
ntrval2	|- ((J e. Top /\ S (_ X) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))

Proof of Theorem ntrval2

Step	Hyp	Ref	Expression
1		difss 2167	. . . . . 6 |- (X \ S) (_ X
2		clscld.1	. . . . . . 7 |- X = U.J
3	2	clsval2 7685	. . . . . 6 |- ((J e. Top /\ (X \ S) (_ X) -> ((cls` J)` (X \ S)) = (X \ ((int` J)` (X \ (X \ S)))))
4	1, 3	mpan2 696	. . . . 5 |- (J e. Top -> ((cls` J)` (X \ S)) = (X \ ((int` J)` (X \ (X \ S)))))
5	4	adantr 389	. . . 4 |- ((J e. Top /\ S (_ X) -> ((cls` J)` (X \ S)) = (X \ ((int` J)` (X \ (X \ S)))))
6		dfss4 2242	. . . . . . . 8 |- (S (_ X <-> (X \ (X \ S)) = S)
7	6	biimp 151	. . . . . . 7 |- (S (_ X -> (X \ (X \ S)) = S)
8	7	fveq2d 3728	. . . . . 6 |- (S (_ X -> ((int` J)` (X \ (X \ S))) = ((int` J)` S))
9	8	adantl 388	. . . . 5 |- ((J e. Top /\ S (_ X) -> ((int` J)` (X \ (X \ S))) = ((int` J)` S))
10	9	difeq2d 2159	. . . 4 |- ((J e. Top /\ S (_ X) -> (X \ ((int` J)` (X \ (X \ S)))) = (X \ ((int` J)` S)))
11	5, 10	eqtrd 1507	. . 3 |- ((J e. Top /\ S (_ X) -> ((cls` J)` (X \ S)) = (X \ ((int` J)` S)))
12	11	difeq2d 2159	. 2 |- ((J e. Top /\ S (_ X) -> (X \ ((cls` J)` (X \ S))) = (X \ (X \ ((int` J)` S))))
13	2	ntropn 7684	. . . 4 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) e. J)
14	2	eltopss 7603	. . . 4 |- ((J e. Top /\ ((int` J)` S) e. J) -> ((int` J)` S) (_ X)
15	13, 14	syldan 467	. . 3 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) (_ X)
16		dfss4 2242	. . 3 |- (((int` J)` S) (_ X <-> (X \ (X \ ((int` J)` S))) = ((int` J)` S))
17	15, 16	sylib 198	. 2 |- ((J e. Top /\ S (_ X) -> (X \ (X \ ((int` J)` S))) = ((int` J)` S))
18	12, 17	eqtr2d 1508	1 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958   \ cdif 2044   (_ wss 2047  U.cuni 2503  ` cfv 3182  Topctop 7588  intcnt 7661  clsccl 7662

This theorem is referenced by:  ntrss 7688  cmntrcld 7694

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-int 2534  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-top 7592  df-cld 7663  df-ntr 7664  df-cls 7665

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