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Theorem 0ded 25757
Description: A deductive system with no object and no morphism. (Contributed by FL, 10-Jan-2008.)
Assertion
Ref Expression
0ded  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded

Proof of Theorem 0ded
Dummy variables  f 
a  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0alg 25756 . . 3  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
2 noel 3459 . . . . . 6  |-  -.  a  e.  (/)
32pm2.21i 123 . . . . 5  |-  ( a  e.  (/)  ->  ( ( (/) `  ( (/) `  a ) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a ) )
4 dm0 4892 . . . . 5  |-  dom  (/)  =  (/)
53, 4eleq2s 2375 . . . 4  |-  ( a  e.  dom  (/)  ->  (
( (/) `  ( (/) `  a ) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a ) )
65rgen 2608 . . 3  |-  A. a  e.  dom  (/) ( ( (/) `  ( (/) `  a ) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a )
7 noel 3459 . . . . . 6  |-  -.  f  e.  (/)
87pm2.21i 123 . . . . 5  |-  ( f  e.  (/)  ->  A. g  e.  dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) ) )
98, 4eleq2s 2375 . . . 4  |-  ( f  e.  dom  (/)  ->  A. g  e.  dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) ) )
109rgen 2608 . . 3  |-  A. f  e.  dom  (/) A. g  e. 
dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) )
111, 6, 103pm3.2i 1130 . 2  |-  ( <. <.
(/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Alg  /\  A. a  e.  dom  (/) ( (
(/) `  ( (/) `  a
) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a )  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) ) )
127pm2.21i 123 . . . . 5  |-  ( f  e.  (/)  ->  A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  f
) ) )
1312, 4eleq2s 2375 . . . 4  |-  ( f  e.  dom  (/)  ->  A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  f
) ) )
1413rgen 2608 . . 3  |-  A. f  e.  dom  (/) A. g  e. 
dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  f
) )
157pm2.21i 123 . . . . 5  |-  ( f  e.  (/)  ->  A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  g
) ) )
1615, 4eleq2s 2375 . . . 4  |-  ( f  e.  dom  (/)  ->  A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  g
) ) )
1716rgen 2608 . . 3  |-  A. f  e.  dom  (/) A. g  e. 
dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  g
) )
1814, 17pm3.2i 441 . 2  |-  ( A. f  e.  dom  (/) A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  f
) )  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  g
) ) )
19 0ex 4150 . . . 4  |-  (/)  e.  _V
2019, 19, 193pm3.2i 1130 . . 3  |-  ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )
21 eqid 2283 . . . 4  |-  dom  (/)  =  dom  (/)
2221, 21isded 25736 . . 3  |-  ( ( ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )  /\  (/)  e.  _V )  ->  ( <. <. (/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Ded  <->  ( ( <. <. (/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Alg  /\  A. a  e.  dom  (/) ( (
(/) `  ( (/) `  a
) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a )  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) ) )  /\  ( A. f  e.  dom  (/)
A. g  e.  dom  (/) ( ( (/) `  g
)  =  ( (/) `  f )  ->  ( (/) `  ( g (/) f ) )  =  ( (/) `  f ) )  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  g
) ) ) ) ) )
2320, 19, 22mp2an 653 . 2  |-  ( <. <.
(/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Ded  <->  ( ( <. <. (/) ,  (/) >. ,  <. (/)
,  (/) >. >.  e.  Alg  /\  A. a  e.  dom  (/) ( (
(/) `  ( (/) `  a
) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a )  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) ) )  /\  ( A. f  e.  dom  (/)
A. g  e.  dom  (/) ( ( (/) `  g
)  =  ( (/) `  f )  ->  ( (/) `  ( g (/) f ) )  =  ( (/) `  f ) )  /\  A. f  e.  dom  (/) A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  g
) ) ) ) )
2411, 18, 23mpbir2an 886 1  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Ded
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    Alg calg 25711   Dedcded 25734
This theorem is referenced by:  0catOLD  25758
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
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