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Theorem hoeq 22340
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 5389 . 2  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
2 ffn 5389 . 2  |-  ( U : ~H --> ~H  ->  U  Fn  ~H )
3 eqfnfv 5622 . . 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 1623   A.wral 2543    Fn wfn 5250   -->wf 5251   ` cfv 5255   ~Hchil 21499
This theorem is referenced by:  hoeqi  22341  homulid2  22380  homco1  22381  homulass  22382  hoadddi  22383  hoadddir  22384  homco2  22557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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