Theorem elunopt 9799

Description: Property defining a unitary Hilbert space operator.

Assertion

Ref	Expression
elunopt	|- (T e. UniOp <-> (T:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y)))

Distinct variable group:   x,y,T

Proof of Theorem elunopt

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (T e. UniOp -> T e. V)
2		fof 3672	. . . 4 |- (T:H~-onto->H~ -> T:H~-->H~)
3		ax-hilex 8869	. . . . 5 |- H~ e. V
4		fex 3652	. . . . 5 |- ((T:H~-->H~ /\ H~ e. V) -> T e. V)
5	3, 4	mpan2 696	. . . 4 |- (T:H~-->H~ -> T e. V)
6	2, 5	syl 10	. . 3 |- (T:H~-onto->H~ -> T e. V)
7	6	adantr 389	. 2 |- ((T:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y)) -> T e. V)
8		foeq1 3668	. . . 4 |- (t = T -> (t:H~-onto->H~ <-> T:H~-onto->H~))
9		fveq1 3723	. . . . . . 7 |- (t = T -> (t` x) = (T` x))
10		fveq1 3723	. . . . . . 7 |- (t = T -> (t` y) = (T` y))
11	9, 10	opreq12d 3978	. . . . . 6 |- (t = T -> ((t` x) .ih (t` y)) = ((T` x) .ih (T` y)))
12	11	eqeq1d 1483	. . . . 5 |- (t = T -> (((t` x) .ih (t` y)) = (x .ih y) <-> ((T` x) .ih (T` y)) = (x .ih y)))
13	12	2ralbidv 1680	. . . 4 |- (t = T -> (A.x e. H~ A.y e. H~ ((t` x) .ih (t` y)) = (x .ih y) <-> A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y)))
14	8, 13	anbi12d 628	. . 3 |- (t = T -> ((t:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((t` x) .ih (t` y)) = (x .ih y)) <-> (T:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y))))
15		df-unop 9769	. . 3 |- UniOp = {t | (t:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((t` x) .ih (t` y)) = (x .ih y))}
16	14, 15	elab2g 1900	. 2 |- (T e. V -> (T e. UniOp <-> (T:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y))))
17	1, 7, 16	pm5.21nii 679	1 |- (T e. UniOp <-> (T:H~-onto->H~ /\ A.x e. H~ A.y e. H~ ((T` x) .ih (T` y)) = (x .ih y)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963  H~chil 8788   .ih csp 8793  UniOpcuo 8818

This theorem is referenced by:  unopt 9839  unopf1ot 9840  cnvunopt 9842  counopt 9845  idunop 9902  lnopuni 9937  elunop2t 9938

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-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-fo 3196  df-fv 3198  df-opr 3965  df-unop 9769

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