Theorem subsp 10465

Description: The subspace topology induced by the topology J and the set A. To Norm: there is no reason to consider that J e. Top and A e. V should be premises. I will transform them into antecedents as soon as possible.

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 10464	. . . 4 |- subSp = {<.<.x, y>., z>. | (y e. Top /\ z = {u | E.v e. y u = (v i^i x)})}
2		visset 1809	. . . . . 6 |- x e. V
3		ibar 642	. . . . . . 7 |- (x e. V -> (y e. Top <-> (x e. V /\ y e. Top)))
4	3	anbi1d 616	. . . . . 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 3988	. . . 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 1492	. . 3 |- subSp = {<.<.x, y>., z>. | ((x e. V /\ y e. Top) /\ z = {u | E.v e. y u = (v i^i x)})}
8		opreq 3958	. . 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 3956	. 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 2227	. . . . . . . . . . 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 2091	. . . . . . . . 9 |- (u = (v i^i A) -> u (_ A)
17	16	pm4.71ri 637	. . . . . . . 8 |- (u = (v i^i A) <-> (u (_ A /\ u = (v i^i A)))
18	17	rexbii 1665	. . . . . . 7 |- (E.v e. J u = (v i^i A) <-> E.v e. J (u (_ A /\ u = (v i^i A)))
19		r19.42v 1761	. . . . . . 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 1572	. . . . 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 2742	. . . . . 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 1541	. . . 4 |- {u | E.v e. J u = (v i^i A)} e. V
25		ineq2 2207	. . . . . . 7 |- (x = A -> (v i^i x) = (v i^i A))
26	25	eqeq2d 1483	. . . . . 6 |- (x = A -> (u = (v i^i x) <-> u = (v i^i A)))
27	26	rexbidv 1661	. . . . 5 |- (x = A -> (E.v e. y u = (v i^i x) <-> E.v e. y u = (v i^i A)))
28	27	abbidv 1574	. . . 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 1784	. . . . 5 |- (y = J -> (E.v e. y u = (v i^i A) <-> E.v e. J u = (v i^i A)))
30	29	abbidv 1574	. . . 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 1473	. . . 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 4019	. . 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 696	. 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 1500	1 |- (subSp` <.A, J>.) = {u | E.v e. J u = (v i^i A)}

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  E.wrex 1643  Vcvv 1807   i^i cin 2042   (_ wss 2043  <.cop 2407  ` cfv 3177  (class class class)co 3954  {copab2 3955  Topctop 7538  subSpcsubsp 10463

This theorem is referenced by:  stoi 10519

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-opr 3956  df-oprab 3957  df-subsp 10464

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