Theorem dfbdop2 9786

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 2208	. . 3 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
2		dfrab2 2274	. . 3 |- {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)} = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
3	1, 2	eqtr4 1498	. 2 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
4		df-bdop 9768	. 2 |- BndLinOp = (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)})
5		lnopft 9785	. . . 4 |- (t e. LinOp -> t:H~-->H~)
6	5	biantrurd 727	. . 3 |- (t e. LinOp -> ((normop` t) < +oo <-> (t:H~-->H~ /\ (normop` t) < +oo)))
7	6	rabbii 1805	. 2 |- {t e. LinOp | (normop` t) < +oo} = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
8	3, 4, 7	3eqtr4 1505	1 |- BndLinOp = {t e. LinOp | (normop` t) < +oo}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  {crab 1648   i^i cin 2046   class class class wbr 2619  -->wf 3178  ` cfv 3182   +oocpnf 5483   < clt 5486  H~chil 8788  norm_opcnop 8814  LinOpclo 8816  BndLinOpcbo 8817

This theorem is referenced by:  elbdopt 9787  hhblo 9828

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-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-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  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-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-lnop 9767  df-bdop 9768

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