Theorem restsn 20974

Description: The only subspace topology induced by the topology { (/) }. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Assertion

Ref	Expression
restsn	|- ( A e. V -> ( { (/) } ↾t A ) = { (/) } )

Proof of Theorem restsn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sn0top 20803	. . . 4 |- { (/) } e. Top
2		elrest 16088	. . . 4 |- ( ( { (/) } e. Top /\ A e. V ) -> ( x e. ( { (/) } ↾t A ) <-> E. y e. { (/) } x = ( y i^i A ) ) )
3	1, 2	mpan 706	. . 3 |- ( A e. V -> ( x e. ( { (/) } ↾t A ) <-> E. y e. { (/) } x = ( y i^i A ) ) )
4		0ex 4790	. . . . 5 |- (/) e. _V
5		ineq1 3807	. . . . . . 7 |- ( y = (/) -> ( y i^i A ) = ( (/) i^i A ) )
6		0in 3969	. . . . . . 7 |- ( (/) i^i A ) = (/)
7	5, 6	syl6eq 2672	. . . . . 6 |- ( y = (/) -> ( y i^i A ) = (/) )
8	7	eqeq2d 2632	. . . . 5 |- ( y = (/) -> ( x = ( y i^i A ) <-> x = (/) ) )
9	4, 8	rexsn 4223	. . . 4 |- ( E. y e. { (/) } x = ( y i^i A ) <-> x = (/) )
10		velsn 4193	. . . 4 |- ( x e. { (/) } <-> x = (/) )
11	9, 10	bitr4i 267	. . 3 |- ( E. y e. { (/) } x = ( y i^i A ) <-> x e. { (/) } )
12	3, 11	syl6bb 276	. 2 |- ( A e. V -> ( x e. ( { (/) } ↾t A ) <-> x e. { (/) } ) )
13	12	eqrdv 2620	1 |- ( A e. V -> ( { (/) } ↾t A ) = { (/) } )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   (/)c0 3915   {csn 4177  (class class class)co 6650   ↾_t crest 16081   Topctop 20698

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rest 16083  df-top 20699  df-topon 20716

This theorem is referenced by: (None)