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Theorem 0catOLD 25758
Description: Category  0 has no object and no morphism. (Contributed by FL, 10-Jan-2008.)
Assertion
Ref Expression
0catOLD  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Cat OLD

Proof of Theorem 0catOLD
Dummy variables  f 
a  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ded 25757 . . 3  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded
2 noel 3459 . . . . . 6  |-  -.  f  e.  (/)
32pm2.21i 123 . . . . 5  |-  ( f  e.  (/)  ->  A. g  e.  dom  (/) A. h  e. 
dom  (/) ( ( (
(/) `  h )  =  ( (/) `  g
)  /\  ( (/) `  g
)  =  ( (/) `  f ) )  -> 
( h (/) ( g
(/) f ) )  =  ( ( h
(/) g ) (/) f ) ) )
4 dm0 4892 . . . . 5  |-  dom  (/)  =  (/)
53, 4eleq2s 2375 . . . 4  |-  ( f  e.  dom  (/)  ->  A. g  e.  dom  (/) A. h  e. 
dom  (/) ( ( (
(/) `  h )  =  ( (/) `  g
)  /\  ( (/) `  g
)  =  ( (/) `  f ) )  -> 
( h (/) ( g
(/) f ) )  =  ( ( h
(/) g ) (/) f ) ) )
65rgen 2608 . . 3  |-  A. f  e.  dom  (/) A. g  e. 
dom  (/) A. h  e. 
dom  (/) ( ( (
(/) `  h )  =  ( (/) `  g
)  /\  ( (/) `  g
)  =  ( (/) `  f ) )  -> 
( h (/) ( g
(/) f ) )  =  ( ( h
(/) g ) (/) f ) )
71, 6pm3.2i 441 . 2  |-  ( <. <.
(/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Ded  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) A. h  e. 
dom  (/) ( ( (
(/) `  h )  =  ( (/) `  g
)  /\  ( (/) `  g
)  =  ( (/) `  f ) )  -> 
( h (/) ( g
(/) f ) )  =  ( ( h
(/) g ) (/) f ) ) )
8 noel 3459 . . . . . 6  |-  -.  a  e.  (/)
98pm2.21i 123 . . . . 5  |-  ( a  e.  (/)  ->  A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( ( (/) `  a ) (/) f )  =  f ) )
109, 4eleq2s 2375 . . . 4  |-  ( a  e.  dom  (/)  ->  A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( ( (/) `  a ) (/) f )  =  f ) )
1110rgen 2608 . . 3  |-  A. a  e.  dom  (/) A. f  e. 
dom  (/) ( ( (/) `  f )  =  a  ->  ( ( (/) `  a ) (/) f )  =  f )
128pm2.21i 123 . . . . 5  |-  ( a  e.  (/)  ->  A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( f (/) ( (/) `  a ) )  =  f ) )
1312, 4eleq2s 2375 . . . 4  |-  ( a  e.  dom  (/)  ->  A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( f (/) ( (/) `  a ) )  =  f ) )
1413rgen 2608 . . 3  |-  A. a  e.  dom  (/) A. f  e. 
dom  (/) ( ( (/) `  f )  =  a  ->  ( f (/) ( (/) `  a ) )  =  f )
1511, 14pm3.2i 441 . 2  |-  ( A. a  e.  dom  (/) A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( ( (/) `  a ) (/) f )  =  f )  /\  A. a  e.  dom  (/) A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( f (/) ( (/) `  a ) )  =  f ) )
16 0ex 4150 . . . 4  |-  (/)  e.  _V
1716, 16, 163pm3.2i 1130 . . 3  |-  ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )
18 eqid 2283 . . . 4  |-  dom  (/)  =  dom  (/)
1918, 18iscatOLD 25754 . . 3  |-  ( ( ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )  /\  (/)  e.  _V )  ->  ( <. <. (/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Cat OLD  <->  (
( <. <. (/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Ded  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) A. h  e. 
dom  (/) ( ( (
(/) `  h )  =  ( (/) `  g
)  /\  ( (/) `  g
)  =  ( (/) `  f ) )  -> 
( h (/) ( g
(/) f ) )  =  ( ( h
(/) g ) (/) f ) ) )  /\  ( A. a  e.  dom  (/) A. f  e. 
dom  (/) ( ( (/) `  f )  =  a  ->  ( ( (/) `  a ) (/) f )  =  f )  /\  A. a  e.  dom  (/) A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( f (/) ( (/) `  a ) )  =  f ) ) ) ) )
2017, 16, 19mp2an 653 . 2  |-  ( <. <.
(/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Cat OLD  <->  (
( <. <. (/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Ded  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) A. h  e. 
dom  (/) ( ( (
(/) `  h )  =  ( (/) `  g
)  /\  ( (/) `  g
)  =  ( (/) `  f ) )  -> 
( h (/) ( g
(/) f ) )  =  ( ( h
(/) g ) (/) f ) ) )  /\  ( A. a  e.  dom  (/) A. f  e. 
dom  (/) ( ( (/) `  f )  =  a  ->  ( ( (/) `  a ) (/) f )  =  f )  /\  A. a  e.  dom  (/) A. f  e.  dom  (/) ( ( (/) `  f )  =  a  ->  ( f (/) ( (/) `  a ) )  =  f ) ) ) )
217, 15, 20mpbir2an 886 1  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Cat OLD
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   (/)c0 3455   <.cop 3643   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Dedcded 25734    Cat OLD ccatOLD 25752
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-uni 3828  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-alg 25716  df-ded 25735  df-catOLD 25753
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