Theorem subspemp 14903

Description: The subspace topology induced by the topology J on the empty set.

Hypothesis

Ref	Expression
subspemp.1	|- J e. Top

Assertion

Ref	Expression
subspemp	|- (subSp` <.(/), J>.) = {(/)}

Proof of Theorem subspemp

Step	Hyp	Ref	Expression
1		in0 2897	. . . . . . 7 |- (y i^i (/)) = (/)
2		eqtr 1904	. . . . . . 7 |- ((x = (y i^i (/)) /\ (y i^i (/)) = (/)) -> x = (/))
3	1, 2	mpan2 760	. . . . . 6 |- (x = (y i^i (/)) -> x = (/))
4	3	a1i 8	. . . . 5 |- (y e. J -> (x = (y i^i (/)) -> x = (/)))
5	4	r19.23aiv 2211	. . . 4 |- (E.y e. J x = (y i^i (/)) -> x = (/))
6		subspemp.1	. . . . . . 7 |- J e. Top
7		0opn 8870	. . . . . . 7 |- (J e. Top -> (/) e. J)
8	6, 7	ax-mp 7	. . . . . 6 |- (/) e. J
9		inidm 2803	. . . . . . 7 |- ((/) i^i (/)) = (/)
10	9	eqcomi 1888	. . . . . 6 |- (/) = ((/) i^i (/))
11		ineq1 2789	. . . . . . . 8 |- (y = (/) -> (y i^i (/)) = ((/) i^i (/)))
12	11	eqeq2d 1895	. . . . . . 7 |- (y = (/) -> ((/) = (y i^i (/)) <-> (/) = ((/) i^i (/))))
13	12	rcla4ev 2381	. . . . . 6 |- (((/) e. J /\ (/) = ((/) i^i (/))) -> E.y e. J (/) = (y i^i (/)))
14	8, 10, 13	mp2an 761	. . . . 5 |- E.y e. J (/) = (y i^i (/))
15		eqeq1 1890	. . . . . 6 |- (x = (/) -> (x = (y i^i (/)) <-> (/) = (y i^i (/))))
16	15	rexbidv 2124	. . . . 5 |- (x = (/) -> (E.y e. J x = (y i^i (/)) <-> E.y e. J (/) = (y i^i (/))))
17	14, 16	mpbiri 211	. . . 4 |- (x = (/) -> E.y e. J x = (y i^i (/)))
18	5, 17	impbii 174	. . 3 |- (E.y e. J x = (y i^i (/)) <-> x = (/))
19	18	abbii 2006	. 2 |- {x | E.y e. J x = (y i^i (/))} = {x | x = (/)}
20		0ex 3446	. . 3 |- (/) e. _V
21	6, 20	subsp 10244	. 2 |- (subSp` <.(/), J>.) = {x | E.y e. J x = (y i^i (/))}
22		df-sn 3049	. 2 |- {(/)} = {x | x = (/)}
23	19, 21, 22	3eqtr4i 1921	1 |- (subSp` <.(/), J>.) = {(/)}

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106   i^i cin 2592  (/)c0 2875  {csn 3044  <.cop 3046  ` cfv 3998  Topctop 8857  subSpcsubsp 10242

This theorem is referenced by:  subspemp2 14904

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opr 4886  df-oprab 4887  df-top 8861  df-subsp 10243

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