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Theorem hoeq 23263
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 5591 . 2  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
2 ffn 5591 . 2  |-  ( U : ~H --> ~H  ->  U  Fn  ~H )
3 eqfnfv 5827 . . 3  |-  ( ( T  Fn  ~H  /\  U  Fn  ~H )  ->  ( T  =  U  <->  A. x  e.  ~H  ( T `  x )  =  ( U `  x ) ) )
43bicomd 193 . 2  |-  ( ( T  Fn  ~H  /\  U  Fn  ~H )  ->  ( A. x  e. 
~H  ( T `  x )  =  ( U `  x )  <-> 
T  =  U ) )
51, 2, 4syl2an 464 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 177    /\ wa 359    = wceq 1652   A.wral 2705    Fn wfn 5449   -->wf 5450   ` cfv 5454   ~Hchil 22422
This theorem is referenced by:  hoeqi  23264  homulid2  23303  homco1  23304  homulass  23305  hoadddi  23306  hoadddir  23307  homco2  23480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462
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