Theorem clsint2 15414

Description: The closure of an intersection is a subset of the intersection of the closures.

Hypothesis

Ref	Expression
clsint2.1	|- X = U.J

Assertion

Ref	Expression
clsint2	|- ((J e. Top /\ C C_ ~PX) -> ((cls` J)` |^|C) C_ |^|_c e. C ((cls` J)` c))

Distinct variable groups:   C,c   J,c   X,c

Proof of Theorem clsint2

Step	Hyp	Ref	Expression
1		elssuni 3206	. . . . . . . . . 10 |- (c e. C -> c C_ U.C)
2		sstr2 2623	. . . . . . . . . 10 |- (c C_ U.C -> (U.C C_ X -> c C_ X))
3	1, 2	syl 12	. . . . . . . . 9 |- (c e. C -> (U.C C_ X -> c C_ X))
4	3	adantl 424	. . . . . . . 8 |- ((J e. Top /\ c e. C) -> (U.C C_ X -> c C_ X))
5		clsint2.1	. . . . . . . . . . . 12 |- X = U.J
6	5	clsss 8963	. . . . . . . . . . 11 |- ((J e. Top /\ c C_ X /\ |^|C C_ c) -> ((cls` J)` |^|C) C_ ((cls` J)` c))
7		intss1 3231	. . . . . . . . . . 11 |- (c e. C -> |^|C C_ c)
8	6, 7	syl3an3 1132	. . . . . . . . . 10 |- ((J e. Top /\ c C_ X /\ c e. C) -> ((cls` J)` |^|C) C_ ((cls` J)` c))
9	8	3com23 1074	. . . . . . . . 9 |- ((J e. Top /\ c e. C /\ c C_ X) -> ((cls` J)` |^|C) C_ ((cls` J)` c))
10	9	3expia 1069	. . . . . . . 8 |- ((J e. Top /\ c e. C) -> (c C_ X -> ((cls` J)` |^|C) C_ ((cls` J)` c)))
11	4, 10	syld 30	. . . . . . 7 |- ((J e. Top /\ c e. C) -> (U.C C_ X -> ((cls` J)` |^|C) C_ ((cls` J)` c)))
12	11	ex 402	. . . . . 6 |- (J e. Top -> (c e. C -> (U.C C_ X -> ((cls` J)` |^|C) C_ ((cls` J)` c))))
13	12	com23 36	. . . . 5 |- (J e. Top -> (U.C C_ X -> (c e. C -> ((cls` J)` |^|C) C_ ((cls` J)` c))))
14	13	imp 377	. . . 4 |- ((J e. Top /\ U.C C_ X) -> (c e. C -> ((cls` J)` |^|C) C_ ((cls` J)` c)))
15		sspwuni 3333	. . . 4 |- (C C_ ~PX <-> U.C C_ X)
16	14, 15	sylan2b 501	. . 3 |- ((J e. Top /\ C C_ ~PX) -> (c e. C -> ((cls` J)` |^|C) C_ ((cls` J)` c)))
17	16	r19.21aiv 2175	. 2 |- ((J e. Top /\ C C_ ~PX) -> A.c e. C ((cls` J)` |^|C) C_ ((cls` J)` c))
18		ssiin 3302	. 2 |- (((cls` J)` |^|C) C_ |^|_c e. C ((cls` J)` c) <-> A.c e. C ((cls` J)` |^|C) C_ ((cls` J)` c))
19	17, 18	sylibr 217	1 |- ((J e. Top /\ C C_ ~PX) -> ((cls` J)` |^|C) C_ |^|_c e. C ((cls` J)` c))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  |^|cint 3214  |^|_ciin 3256  ` cfv 3998  Topctop 8857  clsccl 8938

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-3an 860  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-iin 3258  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-cls 8941

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