Theorem dfbdop2 9781

Description: Alternate definition of the set of bounded linear Hilbert space operators.

Assertion

Ref	Expression
dfbdop2	|- BndLinOp = {t e. LinOp | (normop` t) < +oo}

Proof of Theorem dfbdop2

Step	Hyp	Ref	Expression
1		incom 2211	. . 3 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
2		dfrab2 2277	. . 3 |- {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)} = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
3	1, 2	eqtr4 1501	. 2 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
4		df-bdop 9763	. 2 |- BndLinOp = (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)})
5		lnopft 9780	. . . 4 |- (t e. LinOp -> t:H~-->H~)
6	5	biantrurd 729	. . 3 |- (t e. LinOp -> ((normop` t) < +oo <-> (t:H~-->H~ /\ (normop` t) < +oo)))
7	6	rabbii 1808	. 2 |- {t e. LinOp | (normop` t) < +oo} = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
8	3, 4, 7	3eqtr4 1508	1 |- BndLinOp = {t e. LinOp | (normop` t) < +oo}

Syntax hints:   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  {crab 1651   i^i cin 2049   class class class wbr 2624  -->wf 3184  ` cfv 3188   +oocpnf 5495   < clt 5498  H~chil 8783  norm_opcnop 8809  LinOpclo 8811  BndLinOpcbo 8812

This theorem is referenced by:  elbdopt 9782  hhblo 9823

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-lnop 9762  df-bdop 9763

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