Theorem clsss 7666

Description: Subset relationship for closure.

Hypothesis

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

Assertion

Ref	Expression
clsss	|- ((J e. Top /\ S (_ X /\ T (_ S) -> ((cls` J)` T) (_ ((cls` J)` S))

Proof of Theorem clsss

Step	Hyp	Ref	Expression
1		sstr2 2069	. . . . . 6 |- (T (_ S -> (S (_ x -> T (_ x))
2	1	adantr 389	. . . . 5 |- ((T (_ S /\ x e. (Clsd` J)) -> (S (_ x -> T (_ x))
3	2	ss2rabdv 2125	. . . 4 |- (T (_ S -> {x e. (Clsd` J) | S (_ x} (_ {x e. (Clsd` J) | T (_ x})
4		intss 2551	. . . 4 |- ({x e. (Clsd` J) | S (_ x} (_ {x e. (Clsd` J) | T (_ x} -> |^|{x e. (Clsd` J) | T (_ x} (_ |^|{x e. (Clsd` J) | S (_ x})
5	3, 4	syl 10	. . 3 |- (T (_ S -> |^|{x e. (Clsd` J) | T (_ x} (_ |^|{x e. (Clsd` J) | S (_ x})
6	5	3ad2ant3 801	. 2 |- ((J e. Top /\ S (_ X /\ T (_ S) -> |^|{x e. (Clsd` J) | T (_ x} (_ |^|{x e. (Clsd` J) | S (_ x})
7		clscld.1	. . . 4 |- X = U.J
8	7	clsval 7656	. . 3 |- ((J e. Top /\ T (_ X) -> ((cls` J)` T) = |^|{x e. (Clsd` J) | T (_ x})
9		3simp1 787	. . 3 |- ((J e. Top /\ S (_ X /\ T (_ S) -> J e. Top)
10		sstr2 2069	. . . . 5 |- (T (_ S -> (S (_ X -> T (_ X))
11	10	impcom 351	. . . 4 |- ((S (_ X /\ T (_ S) -> T (_ X)
12	11	3adant1 796	. . 3 |- ((J e. Top /\ S (_ X /\ T (_ S) -> T (_ X)
13	8, 9, 12	sylanc 471	. 2 |- ((J e. Top /\ S (_ X /\ T (_ S) -> ((cls` J)` T) = |^|{x e. (Clsd` J) | T (_ x})
14	7	clsval 7656	. . 3 |- ((J e. Top /\ S (_ X) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S (_ x})
15	14	3adant3 798	. 2 |- ((J e. Top /\ S (_ X /\ T (_ S) -> ((cls` J)` S) = |^|{x e. (Clsd` J) | S (_ x})
16	6, 13, 15	3sstr4d 2102	1 |- ((J e. Top /\ S (_ X /\ T (_ S) -> ((cls` J)` T) (_ ((cls` J)` S))

Syntax hints:   -> wi 3   /\ w3a 774   = wceq 955   e. wcel 957  {crab 1647   (_ wss 2045  U.cuni 2500  |^|cint 2530  ` cfv 3179  Topctop 7567  Clsdccld 7639  clsccl 7641

This theorem is referenced by:  ntrss 7667  clsss2 7682  lpval 7722  lpsscls 7724  dnsconst 7767

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-int 2531  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-fv 3195  df-top 7571  df-cld 7642  df-cls 7644

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