Theorem hoeqt 9626

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 3788	. . 3 |- ((T Fn H~ /\ U Fn H~) -> (T = U <-> (H~ = H~ /\ A.x e. H~ (T` x) = (U` x))))
2		eqid 1473	. . . 4 |- H~ = H~
3	2	biantrur 724	. . 3 |- (A.x e. H~ (T` x) = (U` x) <-> (H~ = H~ /\ A.x e. H~ (T` x) = (U` x)))
4	1, 3	syl6rbbr 538	. 2 |- ((T Fn H~ /\ U Fn H~) -> (A.x e. H~ (T` x) = (U` x) <-> T = U))
5		ffn 3619	. 2 |- (T:H~-->H~ -> T Fn H~)
6		ffn 3619	. 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 954  A.wral 1642   Fn wfn 3172  -->wf 3173  ` cfv 3177  H~chil 8727

This theorem is referenced by:  hoeq 9627  homulid2t 9666  homco1t 9667  homulasst 9668  hoadddit 9669  hoadddirt 9670  homco2t 9840

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-sep 2698  ax-pow 2737  ax-pr 2774

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-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

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