Theorem dfbdop2 9810

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 2219	. . 3 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
2		dfrab2 2285	. . 3 |- {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)} = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
3	1, 2	eqtr4i 1505	. 2 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
4		df-bdop 9792	. 2 |- BndLinOp = (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)})
5		lnopf 9809	. . . 4 |- (t e. LinOp -> t:H~-->H~)
6	5	biantrurd 731	. . 3 |- (t e. LinOp -> ((normop` t) < +oo <-> (t:H~-->H~ /\ (normop` t) < +oo)))
7	6	rabbii 1812	. 2 |- {t e. LinOp | (normop` t) < +oo} = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
8	3, 4, 7	3eqtr4i 1512	1 |- BndLinOp = {t e. LinOp | (normop` t) < +oo}

Syntax hints:   /\ wa 223   = wceq 960   e. wcel 962  {cab 1469  {crab 1655   i^i cin 2057   class class class wbr 2634  -->wf 3194  ` cfv 3198   +oocpnf 5503   < clt 5506  H~chil 8812  norm_opcnop 8838  LinOpclo 8840  BndLinOpcbo 8841

This theorem is referenced by:  elbdop 9811  hhbloi 9852

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-rep 2708  ax-sep 2718  ax-pow 2758  ax-pr 2795  ax-un 2882  ax-hilex 8893

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-rab 1659  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-f 3210  df-fv 3214  df-opr 3981  df-lnop 9791  df-bdop 9792

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