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Theorem 0alg 25859
Description: Lemma for 0ded 25860. (Contributed by FL, 10-Jan-2008.)
Assertion
Ref Expression
0alg  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg

Proof of Theorem 0alg
StepHypRef Expression
1 f0 5441 . . 3  |-  (/) : (/) --> (/)
21, 1, 13pm3.2i 1130 . 2  |-  ( (/) :
(/) --> (/)  /\  (/) : (/) --> (/)  /\  (/) : (/) --> (/) )
3 fun0 5323 . . 3  |-  Fun  (/)
4 ssid 3210 . . . 4  |-  (/)  C_  (/)
5 dm0 4908 . . . 4  |-  dom  (/)  =  (/)
6 xp0r 4784 . . . 4  |-  ( (/)  X.  (/) )  =  (/)
74, 5, 63sstr4i 3230 . . 3  |-  dom  (/)  C_  ( (/) 
X.  (/) )
8 rn0 4952 . . . 4  |-  ran  (/)  =  (/)
98eqimssi 3245 . . 3  |-  ran  (/)  C_  (/)
103, 7, 93pm3.2i 1130 . 2  |-  ( Fun  (/)  /\  dom  (/)  C_  ( (/) 
X.  (/) )  /\  ran  (/)  C_  (/) )
11 0ex 4166 . . . 4  |-  (/)  e.  _V
1211, 11, 113pm3.2i 1130 . . 3  |-  ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )
135eqcomi 2300 . . . 4  |-  (/)  =  dom  (/)
1413, 13isalg 25824 . . 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 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   <.cop 3656    X. cxp 4703   dom cdm 4705   ran crn 4706   Fun wfun 5265   -->wf 5267    Alg calg 25814
This theorem is referenced by:  0ded  25860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-alg 25819
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