Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iri Unicode version

Theorem iri 25800
Description: Composite of an identity with itself. JFM CAT1 th. 59 (Contributed by FL, 5-Dec-2007.)
Hypotheses
Ref Expression
iri.1  |-  O  =  dom  ( id_ `  T
)
iri.2  |-  J  =  ( id_ `  T
)
iri.3  |-  R  =  ( o_ `  T
)
Assertion
Ref Expression
iri  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( J `
 A ) R ( J `  A
) )  =  ( J `  A ) )

Proof of Theorem iri
StepHypRef Expression
1 eqid 2283 . . 3  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
2 iri.1 . . 3  |-  O  =  dom  ( id_ `  T
)
3 iri.2 . . 3  |-  J  =  ( id_ `  T
)
41, 2, 3jdmo 25778 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  dom  ( dom_ `  T )
)
53eqcomi 2287 . . . . . 6  |-  ( id_ `  T )  =  J
65dmeqi 4880 . . . . 5  |-  dom  ( id_ `  T )  =  dom  J
72, 6eqtri 2303 . . . 4  |-  O  =  dom  J
8 eqid 2283 . . . 4  |-  ( dom_ `  T )  =  (
dom_ `  T )
9 eqid 2283 . . . 4  |-  ( cod_ `  T )  =  (
cod_ `  T )
107, 8, 3, 9idosc 25769 . . 3  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( (
dom_ `  T ) `  ( J `  A
) )  =  A  /\  ( ( cod_ `  T ) `  ( J `  A )
)  =  A ) )
1110simprd 449 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( cod_ `  T ) `  ( J `  A )
)  =  A )
12 iri.3 . . . . 5  |-  R  =  ( o_ `  T
)
131, 8, 7, 3, 12, 9cmpida 25774 . . . 4  |-  ( ( T  e.  Cat OLD  /\  A  e.  O  /\  ( J `  A )  e.  dom  ( dom_ `  T ) )  -> 
( ( ( cod_ `  T ) `  ( J `  A )
)  =  A  -> 
( ( J `  A ) R ( J `  A ) )  =  ( J `
 A ) ) )
14133exp 1150 . . 3  |-  ( T  e.  Cat OLD  ->  ( A  e.  O  -> 
( ( J `  A )  e.  dom  ( dom_ `  T )  ->  ( ( ( cod_ `  T ) `  ( J `  A )
)  =  A  -> 
( ( J `  A ) R ( J `  A ) )  =  ( J `
 A ) ) ) ) )
1514imp4b 573 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( ( J `  A )  e.  dom  ( dom_ `  T )  /\  (
( cod_ `  T ) `  ( J `  A
) )  =  A )  ->  ( ( J `  A ) R ( J `  A ) )  =  ( J `  A
) ) )
164, 11, 15mp2and 660 1  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( J `
 A ) R ( J `  A
) )  =  ( J `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   dom cdm 4689   ` cfv 5255  (class class class)co 5858   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715    Cat
OLD ccatOLD 25752
This theorem is referenced by:  immon  25818
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-int 3863  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-alg 25716  df-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735  df-catOLD 25753
  Copyright terms: Public domain W3C validator