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Theorem cmpidb 25878
Description: The 11th "axiom" of a category:  ( J `  A
) is a right neutral element. (Contributed by FL, 24-Oct-2007.)
Hypotheses
Ref Expression
cmpidb.1  |-  M  =  dom  D
cmpidb.2  |-  D  =  ( dom_ `  T
)
cmpidb.3  |-  O  =  dom  J
cmpidb.4  |-  J  =  ( id_ `  T
)
cmpidb.5  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
cmpidb  |-  ( ( T  e.  Cat OLD  /\  A  e.  O  /\  F  e.  M )  ->  ( ( D `  F )  =  A  ->  ( F R ( J `  A
) )  =  F ) )

Proof of Theorem cmpidb
Dummy variables  a 
f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cmpidb.2 . . . 4  |-  D  =  ( dom_ `  T
)
2 eqid 2296 . . . 4  |-  ( cod_ `  T )  =  (
cod_ `  T )
3 cmpidb.4 . . . 4  |-  J  =  ( id_ `  T
)
4 cmpidb.5 . . . 4  |-  R  =  ( o_ `  T
)
5 cmpidb.1 . . . 4  |-  M  =  dom  D
6 cmpidb.3 . . . 4  |-  O  =  dom  J
71, 2, 3, 4, 5, 6cati 25858 . . 3  |-  ( T  e.  Cat OLD  ->  ( ( <. <. D ,  (
cod_ `  T ) >. ,  <. J ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. x  e.  M  A. y  e.  M  (
( ( D `  y )  =  ( ( cod_ `  T
) `  x )  /\  ( D `  x
)  =  ( (
cod_ `  T ) `  f ) )  -> 
( y R ( x R f ) )  =  ( ( y R x ) R f ) ) )  /\  ( A. a  e.  O  A. f  e.  M  (
( ( cod_ `  T
) `  f )  =  a  ->  ( ( J `  a ) R f )  =  f )  /\  A. a  e.  O  A. f  e.  M  (
( D `  f
)  =  a  -> 
( f R ( J `  a ) )  =  f ) ) ) )
8 eqeq2 2305 . . . . . . 7  |-  ( a  =  A  ->  (
( D `  f
)  =  a  <->  ( D `  f )  =  A ) )
9 fveq2 5541 . . . . . . . . 9  |-  ( a  =  A  ->  ( J `  a )  =  ( J `  A ) )
109oveq2d 5890 . . . . . . . 8  |-  ( a  =  A  ->  (
f R ( J `
 a ) )  =  ( f R ( J `  A
) ) )
1110eqeq1d 2304 . . . . . . 7  |-  ( a  =  A  ->  (
( f R ( J `  a ) )  =  f  <->  ( f R ( J `  A ) )  =  f ) )
128, 11imbi12d 311 . . . . . 6  |-  ( a  =  A  ->  (
( ( D `  f )  =  a  ->  ( f R ( J `  a
) )  =  f )  <->  ( ( D `
 f )  =  A  ->  ( f R ( J `  A ) )  =  f ) ) )
13 fveq2 5541 . . . . . . . 8  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
1413eqeq1d 2304 . . . . . . 7  |-  ( f  =  F  ->  (
( D `  f
)  =  A  <->  ( D `  F )  =  A ) )
15 oveq1 5881 . . . . . . . 8  |-  ( f  =  F  ->  (
f R ( J `
 A ) )  =  ( F R ( J `  A
) ) )
16 id 19 . . . . . . . 8  |-  ( f  =  F  ->  f  =  F )
1715, 16eqeq12d 2310 . . . . . . 7  |-  ( f  =  F  ->  (
( f R ( J `  A ) )  =  f  <->  ( F R ( J `  A ) )  =  F ) )
1814, 17imbi12d 311 . . . . . 6  |-  ( f  =  F  ->  (
( ( D `  f )  =  A  ->  ( f R ( J `  A
) )  =  f )  <->  ( ( D `
 F )  =  A  ->  ( F R ( J `  A ) )  =  F ) ) )
1912, 18rspc2v 2903 . . . . 5  |-  ( ( A  e.  O  /\  F  e.  M )  ->  ( A. a  e.  O  A. f  e.  M  ( ( D `
 f )  =  a  ->  ( f R ( J `  a ) )  =  f )  ->  (
( D `  F
)  =  A  -> 
( F R ( J `  A ) )  =  F ) ) )
2019com12 27 . . . 4  |-  ( A. a  e.  O  A. f  e.  M  (
( D `  f
)  =  a  -> 
( f R ( J `  a ) )  =  f )  ->  ( ( A  e.  O  /\  F  e.  M )  ->  (
( D `  F
)  =  A  -> 
( F R ( J `  A ) )  =  F ) ) )
2120ad2antll 709 . . 3  |-  ( ( ( <. <. D ,  (
cod_ `  T ) >. ,  <. J ,  R >. >.  e.  Ded  /\  A. f  e.  M  A. x  e.  M  A. y  e.  M  (
( ( D `  y )  =  ( ( cod_ `  T
) `  x )  /\  ( D `  x
)  =  ( (
cod_ `  T ) `  f ) )  -> 
( y R ( x R f ) )  =  ( ( y R x ) R f ) ) )  /\  ( A. a  e.  O  A. f  e.  M  (
( ( cod_ `  T
) `  f )  =  a  ->  ( ( J `  a ) R f )  =  f )  /\  A. a  e.  O  A. f  e.  M  (
( D `  f
)  =  a  -> 
( f R ( J `  a ) )  =  f ) ) )  ->  (
( A  e.  O  /\  F  e.  M
)  ->  ( ( D `  F )  =  A  ->  ( F R ( J `  A ) )  =  F ) ) )
227, 21syl 15 . 2  |-  ( T  e.  Cat OLD  ->  ( ( A  e.  O  /\  F  e.  M
)  ->  ( ( D `  F )  =  A  ->  ( F R ( J `  A ) )  =  F ) ) )
23223impib 1149 1  |-  ( ( T  e.  Cat OLD  /\  A  e.  O  /\  F  e.  M )  ->  ( ( D `  F )  =  A  ->  ( F R ( J `  A
) )  =  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   dom cdm 4705   ` cfv 5271  (class class class)co 5874   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818   Dedcded 25837    Cat
OLD ccatOLD 25855
This theorem is referenced by:  dualcat2  25887  cmphmib  25902
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528
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-sbc 3005  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-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-catOLD 25856
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