Theorem subsp 10244

Description: The subspace topology induced by the topology J on the set A. (Contributed by FL, 4-Jun-2007.)

Hypotheses

Ref	Expression
subsp.1	|- J e. Top
subsp.2	|- A e. _V

Assertion

Ref	Expression
subsp	|- (subSp` <.A, J>.) = {u | E.v e. J u = (v i^i A)}

Distinct variable groups:   u,A,v   u,J,v

Proof of Theorem subsp

Step	Hyp	Ref	Expression
1		df-subsp 10243	. . . 4 |- subSp = {<.<.x, y>., z>. | (y e. Top /\ z = {u | E.v e. y u = (v i^i x)})}
2		visset 2295	. . . . . 6 |- x e. _V
3		ibar 705	. . . . . . 7 |- (x e. _V -> (y e. Top <-> (x e. _V /\ y e. Top)))
4	3	anbi1d 679	. . . . . 6 |- (x e. _V -> ((y e. Top /\ z = {u | E.v e. y u = (v i^i x)}) <-> ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})))
5	2, 4	ax-mp 7	. . . . 5 |- ((y e. Top /\ z = {u | E.v e. y u = (v i^i x)}) <-> ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)}))
6	5	oprabbii 4923	. . . 4 |- {<.<.x, y>., z>. | (y e. Top /\ z = {u | E.v e. y u = (v i^i x)})} = {<.<.x, y>., z>. | ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}
7	1, 6	eqtri 1908	. . 3 |- subSp = {<.<.x, y>., z>. | ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}
8	7	opreqi 4896	. 2 |- (AsubSpJ) = (A{<.<.x, y>., z>. | ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}J)
9		df-opr 4886	. 2 |- (AsubSpJ) = (subSp` <.A, J>.)
10		subsp.2	. . 3 |- A e. _V
11		subsp.1	. . 3 |- J e. Top
12		id 73	. . . . . . . . . 10 |- (u = (v i^i A) -> u = (v i^i A))
13		inss2 2813	. . . . . . . . . . 11 |- (v i^i A) C_ A
14	13	a1i 8	. . . . . . . . . 10 |- (u = (v i^i A) -> (v i^i A) C_ A)
15	12, 14	eqsstrd 2651	. . . . . . . . 9 |- (u = (v i^i A) -> u C_ A)
16	15	pm4.71ri 700	. . . . . . . 8 |- (u = (v i^i A) <-> (u C_ A /\ u = (v i^i A)))
17	16	rexbii 2128	. . . . . . 7 |- (E.v e. J u = (v i^i A) <-> E.v e. J (u C_ A /\ u = (v i^i A)))
18		r19.42v 2237	. . . . . . 7 |- (E.v e. J (u C_ A /\ u = (v i^i A)) <-> (u C_ A /\ E.v e. J u = (v i^i A)))
19	17, 18	bitri 190	. . . . . 6 |- (E.v e. J u = (v i^i A) <-> (u C_ A /\ E.v e. J u = (v i^i A)))
20	19	abbii 2006	. . . . 5 |- {u | E.v e. J u = (v i^i A)} = {u | (u C_ A /\ E.v e. J u = (v i^i A))}
21		abssexg 3490	. . . . . 6 |- (A e. _V -> {u | (u C_ A /\ E.v e. J u = (v i^i A))} e. _V)
22	10, 21	ax-mp 7	. . . . 5 |- {u | (u C_ A /\ E.v e. J u = (v i^i A))} e. _V
23	20, 22	eqeltri 1967	. . . 4 |- {u | E.v e. J u = (v i^i A)} e. _V
24		ineq2 2790	. . . . . . 7 |- (x = A -> (v i^i x) = (v i^i A))
25	24	eqeq2d 1895	. . . . . 6 |- (x = A -> (u = (v i^i x) <-> u = (v i^i A)))
26	25	rexbidv 2124	. . . . 5 |- (x = A -> (E.v e. y u = (v i^i x) <-> E.v e. y u = (v i^i A)))
27	26	abbidv 2008	. . . 4 |- (x = A -> {u | E.v e. y u = (v i^i x)} = {u | E.v e. y u = (v i^i A)})
28		rexeq 2267	. . . . 5 |- (y = J -> (E.v e. y u = (v i^i A) <-> E.v e. J u = (v i^i A)))
29	28	abbidv 2008	. . . 4 |- (y = J -> {u | E.v e. y u = (v i^i A)} = {u | E.v e. J u = (v i^i A)})
30		eqid 1884	. . . 4 |- {<.<.x, y>., z>. | ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})} = {<.<.x, y>., z>. | ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}
31	23, 27, 29, 30	oprabval2 4957	. . 3 |- ((A e. _V /\ J e. Top) -> (A{<.<.x, y>., z>. | ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}J) = {u | E.v e. J u = (v i^i A)})
32	10, 11, 31	mp2an 761	. 2 |- (A{<.<.x, y>., z>. | ((x e. _V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}J) = {u | E.v e. J u = (v i^i A)}
33	8, 9, 32	3eqtr3i 1918	1 |- (subSp` <.A, J>.) = {u | E.v e. J u = (v i^i A)}

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  <.cop 3046  ` cfv 3998  (class class class)co 4884  {copab2 4885  Topctop 8857  subSpcsubsp 10242

This theorem is referenced by:  issubsp 10245  stoiglem 10250  subsp2 14902  subspemp 14903  stoig2b 14906  stoi 14998

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-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-subsp 10243

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