Theorem indistps2 19272

Description: The indiscrete topology on a set A expressed as a topological space, using direct component assignments. Compare with indistps 19271. The advantage of this version is that it is the shortest to state and easiest to work with in most situations. Theorems indistpsALT 19273 and indistps2ALT 19274 show that the two forms can be derived from each other. (Contributed by NM, 24-Oct-2012.)

Hypotheses

Ref	Expression
indistps2.a	|- ( Base ` K ) = A
indistps2.j	|- ( TopOpen ` K ) = { (/) , A }

Assertion

Ref	Expression
indistps2	|- K e. TopSp

Proof of Theorem indistps2

Step	Hyp	Ref	Expression
1		indistps2.a	. 2 |- ( Base ` K ) = A
2		indistps2.j	. 2 |- ( TopOpen ` K ) = { (/) , A }
3		0ex 4570	. . . 4 |- (/) e. _V
4		fvex 5867	. . . . 5 |- ( Base ` K ) e. _V
5	1, 4	eqeltrri 2545	. . . 4 |- A e. _V
6	3, 5	unipr 4251	. . 3 |- U. { (/) , A } = ( (/) u. A )
7		uncom 3641	. . 3 |- ( (/) u. A ) = ( A u. (/) )
8		un0 3803	. . 3 |- ( A u. (/) ) = A
9	6, 7, 8	3eqtrri 2494	. 2 |- A = U. { (/) , A }
10		indistop 19262	. 2 |- { (/) , A } e. Top
11	1, 2, 9, 10	istpsi 19205	1 |- K e. TopSp

Syntax hints:   = wceq 1374   e. wcel 1762   _Vcvv 3106   u. cun 3467   (/)c0 3778   {cpr 4022   U.cuni 4238   ` cfv 5579   Basecbs 14479   TopOpenctopn 14666   TopSpctps 19157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-top 19159  df-topon 19162  df-topsp 19163

This theorem is referenced by: (None)