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Theorem cmpidb 25775
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 2283 . . . 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 25755 . . 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 2292 . . . . . . 7  |-  ( a  =  A  ->  (
( D `  f
)  =  a  <->  ( D `  f )  =  A ) )
9 fveq2 5525 . . . . . . . . 9  |-  ( a  =  A  ->  ( J `  a )  =  ( J `  A ) )
109oveq2d 5874 . . . . . . . 8  |-  ( a  =  A  ->  (
f R ( J `
 a ) )  =  ( f R ( J `  A
) ) )
1110eqeq1d 2291 . . . . . . 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 5525 . . . . . . . 8  |-  ( f  =  F  ->  ( D `  f )  =  ( D `  F ) )
1413eqeq1d 2291 . . . . . . 7  |-  ( f  =  F  ->  (
( D `  f
)  =  A  <->  ( D `  F )  =  A ) )
15 oveq1 5865 . . . . . . . 8  |-  ( f  =  F  ->  (
f R ( J `
 A ) )  =  ( F R ( J `  A
) ) )
16 id 19 . . . . . . . 8  |-  ( f  =  F  ->  f  =  F )
1715, 16eqeq12d 2297 . . . . . . 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 2890 . . . . 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 1623    e. wcel 1684   A.wral 2543   <.cop 3643   dom cdm 4689   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715   Dedcded 25734    Cat
OLD ccatOLD 25752
This theorem is referenced by:  dualcat2  25784  cmphmib  25799
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-sbc 2992  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-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-catOLD 25753
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