Theorem hoeq 11323

Description: Equality of Hilbert space operators.

Assertion

Ref	Expression
hoeq	|- ((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 hoeq

Step	Hyp	Ref	Expression
1		eqfnfv2 4767	. . 3 |- ((T Fn ~H /\ U Fn ~H) -> (T = U <-> A.x e. ~H (T` x) = (U` x)))
2	1	bicomd 580	. 2 |- ((T Fn ~H /\ U Fn ~H) -> (A.x e. ~H (T` x) = (U` x) <-> T = U))
3		ffn 4562	. 2 |- (T:~H-->~H -> T Fn ~H)
4		ffn 4562	. 2 |- (U:~H-->~H -> U Fn ~H)
5	2, 3, 4	syl2an 503	1 |- ((T:~H-->~H /\ U:~H-->~H) -> (A.x e. ~H (T` x) = (U` x) <-> T = U))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298  A.wral 2105   Fn wfn 3993  -->wf 3994  ` cfv 3998  ~Hchil 10420

This theorem is referenced by:  hoeqi 11324  homulid2 11363  homco1 11364  homulass 11365  hoadddi 11366  hoadddir 11367  homco2 11538

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014

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