Theorem ntrcmp 15406

Description: An interior of a complement is the complement of the closure. This set is also known as the exterior of A.

Hypothesis

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

Assertion

Ref	Expression
ntrcmp	|- ((J e. Top /\ A C_ X) -> ((int` J)` (X \ A)) = (X \ ((cls` J)` A)))

Proof of Theorem ntrcmp

Step	Hyp	Ref	Expression
1		difss 2735	. . . 4 |- (X \ A) C_ X
2		ntrcmp.1	. . . . 5 |- X = U.J
3	2	ntrval2 8962	. . . 4 |- ((J e. Top /\ (X \ A) C_ X) -> ((int` J)` (X \ A)) = (X \ ((cls` J)` (X \ (X \ A)))))
4	1, 3	mpan2 760	. . 3 |- (J e. Top -> ((int` J)` (X \ A)) = (X \ ((cls` J)` (X \ (X \ A)))))
5	4	adantr 425	. 2 |- ((J e. Top /\ A C_ X) -> ((int` J)` (X \ A)) = (X \ ((cls` J)` (X \ (X \ A)))))
6		simpr 350	. . . . 5 |- ((J e. Top /\ A C_ X) -> A C_ X)
7		dfss4 2827	. . . . 5 |- (A C_ X <-> (X \ (X \ A)) = A)
8	6, 7	sylib 215	. . . 4 |- ((J e. Top /\ A C_ X) -> (X \ (X \ A)) = A)
9	8	fveq2d 4685	. . 3 |- ((J e. Top /\ A C_ X) -> ((cls` J)` (X \ (X \ A))) = ((cls` J)` A))
10	9	difeq2d 2726	. 2 |- ((J e. Top /\ A C_ X) -> (X \ ((cls` J)` (X \ (X \ A)))) = (X \ ((cls` J)` A)))
11	5, 10	eqtrd 1925	1 |- ((J e. Top /\ A C_ X) -> ((int` J)` (X \ A)) = (X \ ((cls` J)` A)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   \ cdif 2590   C_ wss 2593  U.cuni 3177  ` cfv 3998  Topctop 8857  intcnt 8937  clsccl 8938

This theorem is referenced by:  clsun 15413  isnrm2 15552

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941

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