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Theorem iri 25903
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 2296 . . 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 25881 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  dom  ( dom_ `  T )
)
53eqcomi 2300 . . . . . 6  |-  ( id_ `  T )  =  J
65dmeqi 4896 . . . . 5  |-  dom  ( id_ `  T )  =  dom  J
72, 6eqtri 2316 . . . 4  |-  O  =  dom  J
8 eqid 2296 . . . 4  |-  ( dom_ `  T )  =  (
dom_ `  T )
9 eqid 2296 . . . 4  |-  ( cod_ `  T )  =  (
cod_ `  T )
107, 8, 3, 9idosc 25872 . . 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 25877 . . . 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 1632    e. wcel 1696   dom cdm 4705   ` cfv 5271  (class class class)co 5874   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818    Cat
OLD ccatOLD 25855
This theorem is referenced by:  immon  25921
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-int 3879  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-alg 25819  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-ded 25838  df-catOLD 25856
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