Theorem dfbdop2 9726

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 2204	. . 3 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
2		dfrab2 2270	. . 3 |- {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)} = ({t | (t:H~-->H~ /\ (normop` t) < +oo)} i^i LinOp)
3	1, 2	eqtr4 1495	. 2 |- (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)}) = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
4		df-bdop 9708	. 2 |- BndLinOp = (LinOp i^i {t | (t:H~-->H~ /\ (normop` t) < +oo)})
5		lnopft 9725	. . . 4 |- (t e. LinOp -> t:H~-->H~)
6	5	biantrurd 726	. . 3 |- (t e. LinOp -> ((normop` t) < +oo <-> (t:H~-->H~ /\ (normop` t) < +oo)))
7	6	rabbii 1801	. 2 |- {t e. LinOp | (normop` t) < +oo} = {t e. LinOp | (t:H~-->H~ /\ (normop` t) < +oo)}
8	3, 4, 7	3eqtr4 1502	1 |- BndLinOp = {t e. LinOp | (normop` t) < +oo}

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  {crab 1645   i^i cin 2042   class class class wbr 2614  -->wf 3173  ` cfv 3177   +oocpnf 5463   < clt 5466  H~chil 8727  norm_opcnop 8753  LinOpclo 8755  BndLinOpcbo 8756

This theorem is referenced by:  elbdopt 9727  hhblo 9768

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-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-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-hilex 8808

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  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-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-lnop 9707  df-bdop 9708

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