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Theorem 0ded 25860
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 25859 . . 3  |-  <. <. (/) ,  (/) >. ,  <. (/) ,  (/) >. >.  e.  Alg
2 noel 3472 . . . . . 6  |-  -.  a  e.  (/)
32pm2.21i 123 . . . . 5  |-  ( a  e.  (/)  ->  ( ( (/) `  ( (/) `  a ) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a ) )
4 dm0 4908 . . . . 5  |-  dom  (/)  =  (/)
53, 4eleq2s 2388 . . . 4  |-  ( a  e.  dom  (/)  ->  (
( (/) `  ( (/) `  a ) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a ) )
65rgen 2621 . . 3  |-  A. a  e.  dom  (/) ( ( (/) `  ( (/) `  a ) )  =  a  /\  ( (/) `  ( (/) `  a ) )  =  a )
7 noel 3472 . . . . . 6  |-  -.  f  e.  (/)
87pm2.21i 123 . . . . 5  |-  ( f  e.  (/)  ->  A. g  e.  dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) ) )
98, 4eleq2s 2388 . . . 4  |-  ( f  e.  dom  (/)  ->  A. g  e.  dom  (/) ( <. g ,  f >.  e.  dom  (/)  <->  (
(/) `  g )  =  ( (/) `  f
) ) )
109rgen 2621 . . 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 2388 . . . 4  |-  ( f  e.  dom  (/)  ->  A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  f
) ) )
1413rgen 2621 . . 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 2388 . . . 4  |-  ( f  e.  dom  (/)  ->  A. g  e.  dom  (/) ( ( (/) `  g )  =  (
(/) `  f )  ->  ( (/) `  ( g
(/) f ) )  =  ( (/) `  g
) ) )
1716rgen 2621 . . 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 4166 . . . 4  |-  (/)  e.  _V
2019, 19, 193pm3.2i 1130 . . 3  |-  ( (/)  e.  _V  /\  (/)  e.  _V  /\  (/)  e.  _V )
21 eqid 2296 . . . 4  |-  dom  (/)  =  dom  (/)
2221, 21isded 25839 . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   (/)c0 3468   <.cop 3656   dom cdm 4705   ` cfv 5271  (class class class)co 5874    Alg calg 25814   Dedcded 25837
This theorem is referenced by:  0catOLD  25861
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-uni 3844  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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-alg 25819  df-ded 25838
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