Theorem hoeqt 9686

Description: Equality of Hilbert space operators.

Assertion

Ref	Expression
hoeqt	|- ((T:H~-->H~ /\ U:H~-->H~) -> (A.x e. H~ (T` x) = (U` x) <-> T = U))

Distinct variable groups:   x,T   x,U

Proof of Theorem hoeqt

Step	Hyp	Ref	Expression
1		eqfnfv 3797	. . 3 |- ((T Fn H~ /\ U Fn H~) -> (T = U <-> (H~ = H~ /\ A.x e. H~ (T` x) = (U` x))))
2		eqid 1475	. . . 4 |- H~ = H~
3	2	biantrur 725	. . 3 |- (A.x e. H~ (T` x) = (U` x) <-> (H~ = H~ /\ A.x e. H~ (T` x) = (U` x)))
4	1, 3	syl6rbbr 539	. 2 |- ((T Fn H~ /\ U Fn H~) -> (A.x e. H~ (T` x) = (U` x) <-> T = U))
5		ffn 3627	. 2 |- (T:H~-->H~ -> T Fn H~)
6		ffn 3627	. 2 |- (U:H~-->H~ -> U Fn H~)
7	4, 5, 6	syl2an 454	1 |- ((T:H~-->H~ /\ U:H~-->H~) -> (A.x e. H~ (T` x) = (U` x) <-> T = U))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956  A.wral 1645   Fn wfn 3177  -->wf 3178  ` cfv 3182  H~chil 8788

This theorem is referenced by:  hoeq 9687  homulid2t 9726  homco1t 9727  homulasst 9728  hoadddit 9729  hoadddirt 9730  homco2t 9901

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-sep 2703  ax-pow 2742  ax-pr 2779

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-fv 3198

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