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Theorem hoeq 22395
Description: Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
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
StepHypRef Expression
1 ffn 5427 . 2  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
2 ffn 5427 . 2  |-  ( U : ~H --> ~H  ->  U  Fn  ~H )
3 eqfnfv 5660 . . 3  |-  ( ( T  Fn  ~H  /\  U  Fn  ~H )  ->  ( T  =  U  <->  A. x  e.  ~H  ( T `  x )  =  ( U `  x ) ) )
43bicomd 192 . 2  |-  ( ( T  Fn  ~H  /\  U  Fn  ~H )  ->  ( A. x  e. 
~H  ( T `  x )  =  ( U `  x )  <-> 
T  =  U ) )
51, 2, 4syl2an 463 1  |-  ( ( T : ~H --> ~H  /\  U : ~H --> ~H )  ->  ( A. x  e. 
~H  ( T `  x )  =  ( U `  x )  <-> 
T  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633   A.wral 2577    Fn wfn 5287   -->wf 5288   ` cfv 5292   ~Hchil 21554
This theorem is referenced by:  hoeqi  22396  homulid2  22435  homco1  22436  homulass  22437  hoadddi  22438  hoadddir  22439  homco2  22612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300
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