Theorem issubsplem1 10246

Description: The predicate "is a subspace topology". (Contributed by FL, 28-Jan-2009.)

Hypothesis

Ref	Expression
issubsplem1.1	|- J e. Top

Assertion

Ref	Expression
issubsplem1	|- ((A e. C /\ B e. _V) -> (A e. (subSp` <.B, J>.) <-> E.v e. J A = (v i^i B)))

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

Proof of Theorem issubsplem1

Step	Hyp	Ref	Expression
1		opeq1 3158	. . . . . . 7 |- (B = if(B e. _V, B, (/)) -> <.B, J>. = <.if(B e. _V, B, (/)), J>.)
2	1	fveq2d 4685	. . . . . 6 |- (B = if(B e. _V, B, (/)) -> (subSp` <.B, J>.) = (subSp` <.if(B e. _V, B, (/)), J>.))
3	2	eleq2d 1964	. . . . 5 |- (B = if(B e. _V, B, (/)) -> (A e. (subSp` <.B, J>.) <-> A e. (subSp` <.if(B e. _V, B, (/)), J>.)))
4		ineq2 2790	. . . . . . 7 |- (B = if(B e. _V, B, (/)) -> (v i^i B) = (v i^i if(B e. _V, B, (/))))
5	4	eqeq2d 1895	. . . . . 6 |- (B = if(B e. _V, B, (/)) -> (A = (v i^i B) <-> A = (v i^i if(B e. _V, B, (/)))))
6	5	rexbidv 2124	. . . . 5 |- (B = if(B e. _V, B, (/)) -> (E.v e. J A = (v i^i B) <-> E.v e. J A = (v i^i if(B e. _V, B, (/)))))
7	3, 6	bibi12d 691	. . . 4 |- (B = if(B e. _V, B, (/)) -> ((A e. (subSp` <.B, J>.) <-> E.v e. J A = (v i^i B)) <-> (A e. (subSp` <.if(B e. _V, B, (/)), J>.) <-> E.v e. J A = (v i^i if(B e. _V, B, (/))))))
8	7	imbi2d 674	. . 3 |- (B = if(B e. _V, B, (/)) -> ((A e. C -> (A e. (subSp` <.B, J>.) <-> E.v e. J A = (v i^i B))) <-> (A e. C -> (A e. (subSp` <.if(B e. _V, B, (/)), J>.) <-> E.v e. J A = (v i^i if(B e. _V, B, (/)))))))
9		0ex 3446	. . . . 5 |- (/) e. _V
10	9	elimel 3025	. . . 4 |- if(B e. _V, B, (/)) e. _V
11		issubsplem1.1	. . . 4 |- J e. Top
12	10, 11	issubsp 10245	. . 3 |- (A e. C -> (A e. (subSp` <.if(B e. _V, B, (/)), J>.) <-> E.v e. J A = (v i^i if(B e. _V, B, (/)))))
13	8, 12	dedth 3011	. 2 |- (B e. _V -> (A e. C -> (A e. (subSp` <.B, J>.) <-> E.v e. J A = (v i^i B))))
14	13	impcom 378	1 |- ((A e. C /\ B e. _V) -> (A e. (subSp` <.B, J>.) <-> E.v e. J A = (v i^i B)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   i^i cin 2592  (/)c0 2875  ifcif 2982  <.cop 3046  ` cfv 3998  Topctop 8857  subSpcsubsp 10242

This theorem is referenced by:  issubspt 10247

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-if 2983  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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