Theorem subsp 10554

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

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 10553	. . . 4 |- subSp = {<.<.x, y>., z>. | (y e. Top /\ z = {u | E.v e. y u = (v i^i x)})}
2		visset 1813	. . . . . 6 |- x e. V
3		ibar 643	. . . . . . 7 |- (x e. V -> (y e. Top <-> (x e. V /\ y e. Top)))
4	3	anbi1d 617	. . . . . 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 3997	. . . 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	eqtr 1495	. . 3 |- subSp = {<.<.x, y>., z>. | ((x e. V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}
8		opreq 3967	. . 3 |- (subSp = {<.<.x, y>., z>. | ((x e. V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})} -> (AsubSpJ) = (A{<.<.x, y>., z>. | ((x e. V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}J))
9	7, 8	ax-mp 7	. 2 |- (AsubSpJ) = (A{<.<.x, y>., z>. | ((x e. V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}J)
10		df-opr 3965	. 2 |- (AsubSpJ) = (subSp` <.A, J>.)
11		subsp.2	. . 3 |- A e. V
12		subsp.1	. . 3 |- J e. Top
13		id 59	. . . . . . . . . 10 |- (u = (v i^i A) -> u = (v i^i A))
14		inss2 2231	. . . . . . . . . . 11 |- (v i^i A) (_ A
15	14	a1i 8	. . . . . . . . . 10 |- (u = (v i^i A) -> (v i^i A) (_ A)
16	13, 15	eqsstrd 2095	. . . . . . . . 9 |- (u = (v i^i A) -> u (_ A)
17	16	pm4.71ri 638	. . . . . . . 8 |- (u = (v i^i A) <-> (u (_ A /\ u = (v i^i A)))
18	17	rexbii 1668	. . . . . . 7 |- (E.v e. J u = (v i^i A) <-> E.v e. J (u (_ A /\ u = (v i^i A)))
19		r19.42v 1764	. . . . . . 7 |- (E.v e. J (u (_ A /\ u = (v i^i A)) <-> (u (_ A /\ E.v e. J u = (v i^i A)))
20	18, 19	bitr 173	. . . . . 6 |- (E.v e. J u = (v i^i A) <-> (u (_ A /\ E.v e. J u = (v i^i A)))
21	20	abbii 1575	. . . . 5 |- {u | E.v e. J u = (v i^i A)} = {u | (u (_ A /\ E.v e. J u = (v i^i A))}
22		abssexg 2747	. . . . . 6 |- (A e. V -> {u | (u (_ A /\ E.v e. J u = (v i^i A))} e. V)
23	11, 22	ax-mp 7	. . . . 5 |- {u | (u (_ A /\ E.v e. J u = (v i^i A))} e. V
24	21, 23	eqeltr 1544	. . . 4 |- {u | E.v e. J u = (v i^i A)} e. V
25		ineq2 2211	. . . . . . 7 |- (x = A -> (v i^i x) = (v i^i A))
26	25	eqeq2d 1486	. . . . . 6 |- (x = A -> (u = (v i^i x) <-> u = (v i^i A)))
27	26	rexbidv 1664	. . . . 5 |- (x = A -> (E.v e. y u = (v i^i x) <-> E.v e. y u = (v i^i A)))
28	27	abbidv 1577	. . . 4 |- (x = A -> {u | E.v e. y u = (v i^i x)} = {u | E.v e. y u = (v i^i A)})
29		rexeq1 1787	. . . . 5 |- (y = J -> (E.v e. y u = (v i^i A) <-> E.v e. J u = (v i^i A)))
30	29	abbidv 1577	. . . 4 |- (y = J -> {u | E.v e. y u = (v i^i A)} = {u | E.v e. J u = (v i^i A)})
31		eqid 1475	. . . 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)})}
32	24, 28, 30, 31	oprabval2 4028	. . 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)})
33	11, 12, 32	mp2an 697	. 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)}
34	9, 10, 33	3eqtr3 1503	1 |- (subSp` <.A, J>.) = {u | E.v e. J u = (v i^i A)}

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646  Vcvv 1811   i^i cin 2046   (_ wss 2047  <.cop 2411  ` cfv 3182  (class class class)co 3963  {copab2 3964  Topctop 7588  subSpcsubsp 10552

This theorem is referenced by:  stoi 10639

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-oprab 3966  df-subsp 10553

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