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Theorem 0alg 25756
Description: Lemma for 0ded 25757. (Contributed by FL, 10-Jan-2008.)
Assertion
Ref Expression
0alg  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg

Proof of Theorem 0alg
StepHypRef Expression
1 f0 5425 . . 3  |-  (/) : (/) --> (/)
21, 1, 13pm3.2i 1130 . 2  |-  ( (/) :
(/) --> (/)  /\  (/) : (/) --> (/)  /\  (/) : (/) --> (/) )
3 fun0 5307 . . 3  |-  Fun  (/)
4 ssid 3197 . . . 4  |-  (/)  C_  (/)
5 dm0 4892 . . . 4  |-  dom  (/)  =  (/)
6 xp0r 4768 . . . 4  |-  ( (/)  X.  (/) )  =  (/)
74, 5, 63sstr4i 3217 . . 3  |-  dom  (/)  C_  ( (/) 
X.  (/) )
8 rn0 4936 . . . 4  |-  ran  (/)  =  (/)
98eqimssi 3232 . . 3  |-  ran  (/)  C_  (/)
103, 7, 93pm3.2i 1130 . 2  |-  ( Fun  (/)  /\  dom  (/)  C_  ( (/) 
X.  (/) )  /\  ran  (/)  C_  (/) )
11 0ex 4150 . . . 4  |-  (/)  e.  _V
1211, 11, 113pm3.2i 1130 . . 3  |-  ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )
135eqcomi 2287 . . . 4  |-  (/)  =  dom  (/)
1413, 13isalg 25721 . . 3  |-  ( ( ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )  /\  (/)  e.  _V )  ->  ( <. <. (/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Alg  <->  ( ( (/)
: (/) --> (/)  /\  (/) : (/) --> (/)  /\  (/) : (/) --> (/) )  /\  ( Fun  (/)  /\  dom  (/)  C_  ( (/) 
X.  (/) )  /\  ran  (/)  C_  (/) ) ) ) )
1512, 11, 14mp2an 653 . 2  |-  ( <. <.
(/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Alg  <->  ( ( (/)
: (/) --> (/)  /\  (/) : (/) --> (/)  /\  (/) : (/) --> (/) )  /\  ( Fun  (/)  /\  dom  (/)  C_  ( (/) 
X.  (/) )  /\  ran  (/)  C_  (/) ) ) )
162, 10, 15mpbir2an 886 1  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   <.cop 3643    X. cxp 4687   dom cdm 4689   ran crn 4690   Fun wfun 5249   -->wf 5251    Alg calg 25711
This theorem is referenced by:  0ded  25757
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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-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-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-alg 25716
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