Theorem indistps2ALT 20818

Description: The indiscrete topology on a set A expressed as a topological space, using direct component assignments. Here we show how to derive the direct component assignment version indistps2 20816 from the structural version indistps 20815. (Contributed by NM, 24-Oct-2012.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
indistps2ALT.a	|- ( Base ` K ) = A
indistps2ALT.j	|- ( TopOpen ` K ) = { (/) , A }

Assertion

Ref	Expression
indistps2ALT	|- K e. TopSp

Proof of Theorem indistps2ALT

Step	Hyp	Ref	Expression
1		indistps2ALT.a	. . . 4 |- ( Base ` K ) = A
2		fvex 6201	. . . 4 |- ( Base ` K ) e. _V
3	1, 2	eqeltrri 2698	. . 3 |- A e. _V
4		indistopon 20805	. . 3 |- ( A e. _V -> { (/) , A } e. (TopOn ` A ) )
5	3, 4	ax-mp 5	. 2 |- { (/) , A } e. (TopOn ` A )
6	1	eqcomi 2631	. . 3 |- A = ( Base ` K )
7		indistps2ALT.j	. . . 4 |- ( TopOpen ` K ) = { (/) , A }
8	7	eqcomi 2631	. . 3 |- { (/) , A } = ( TopOpen ` K )
9	6, 8	istps 20738	. 2 |- ( K e. TopSp <-> { (/) , A } e. (TopOn ` A ) )
10	5, 9	mpbir 221	1 |- K e. TopSp

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {cpr 4179   ` cfv 5888   Basecbs 15857   TopOpenctopn 16082  TopOnctopon 20715   TopSpctps 20736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-top 20699  df-topon 20716  df-topsp 20737

This theorem is referenced by: (None)