Theorem ntrval2 7636

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 2163	. . . . . 6 |- (X \ S) (_ X
2		clscld.1	. . . . . . 7 |- X = U.J
3	2	clsval2 7635	. . . . . 6 |- ((J e. Top /\ (X \ S) (_ X) -> ((cls` J)` (X \ S)) = (X \ ((int` J)` (X \ (X \ S)))))
4	1, 3	mpan2 695	. . . . 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 2238	. . . . . . . 8 |- (S (_ X <-> (X \ (X \ S)) = S)
7	6	biimp 151	. . . . . . 7 |- (S (_ X -> (X \ (X \ S)) = S)
8	7	fveq2d 3719	. . . . . 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 2155	. . . 4 |- ((J e. Top /\ S (_ X) -> (X \ ((int` J)` (X \ (X \ S)))) = (X \ ((int` J)` S)))
11	5, 10	eqtrd 1504	. . 3 |- ((J e. Top /\ S (_ X) -> ((cls` J)` (X \ S)) = (X \ ((int` J)` S)))
12	11	difeq2d 2155	. 2 |- ((J e. Top /\ S (_ X) -> (X \ ((cls` J)` (X \ S))) = (X \ (X \ ((int` J)` S))))
13	2	ntropn 7634	. . . 4 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) e. J)
14	2	eltopss 7553	. . . 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 2238	. . 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 1505	1 |- ((J e. Top /\ S (_ X) -> ((int` J)` S) = (X \ ((cls` J)` (X \ S))))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956   \ cdif 2040   (_ wss 2043  U.cuni 2498  ` cfv 3177  Topctop 7538  intcnt 7611  clsccl 7612

This theorem is referenced by:  ntrss 7638  cmntrcld 7644

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-int 2529  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-top 7542  df-cld 7613  df-ntr 7614  df-cls 7615

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