Theorem ist1 10585

Description: The predicate J is T1.

Hypothesis

Ref	Expression
ist1.1	|- X = U.J

Assertion

Ref	Expression
ist1	|- (J e. T1 <-> (J e. Top /\ A.a e. X {a} e. (Clsd` J)))

Distinct variable group:   J,a

Proof of Theorem ist1

Step	Hyp	Ref	Expression
1		unieq 2514	. . . . . 6 |- (x = J -> U.x = U.J)
2		ist1.1	. . . . . 6 |- X = U.J
3	1, 2	syl6eqr 1528	. . . . 5 |- (x = J -> U.x = X)
4	3	eleq2d 1544	. . . 4 |- (x = J -> (a e. U.x <-> a e. X))
5		fveq2 3730	. . . . 5 |- (x = J -> (Clsd` x) = (Clsd` J))
6	5	eleq2d 1544	. . . 4 |- (x = J -> ({a} e. (Clsd` x) <-> {a} e. (Clsd` J)))
7	4, 6	imbi12d 628	. . 3 |- (x = J -> ((a e. U.x -> {a} e. (Clsd` x)) <-> (a e. X -> {a} e. (Clsd` J))))
8	7	ralbidv2 1668	. 2 |- (x = J -> (A.a e. U.x{a} e. (Clsd` x) <-> A.a e. X {a} e. (Clsd` J)))
9		df-t1 10584	. 2 |- T1 = {x e. Top | A.a e. U.x{a} e. (Clsd` x)}
10	8, 9	elrab2 1910	1 |- (J e. T1 <-> (J e. Top /\ A.a e. X {a} e. (Clsd` J)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  {csn 2413  U.cuni 2507  ` cfv 3188  Topctop 7590  Clsdccld 7657  T1ct1 10582

This theorem is referenced by:  t2t1 10587

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-t1 10584

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