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Theorem homib 25796
Description: The homset which  ( ( id_ `  T ) `
 A ) belongs to. JFM CAT1 th. 55. (Contributed by FL, 5-Dec-2007.)
Hypotheses
Ref Expression
homib.1  |-  O  =  dom  ( id_ `  T
)
homib.2  |-  J  =  ( id_ `  T
)
homib.3  |-  H  =  ( hom `  T
)
Assertion
Ref Expression
homib  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  ( H `  <. A ,  A >. ) )

Proof of Theorem homib
StepHypRef Expression
1 eqid 2283 . . . 4  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
2 homib.1 . . . 4  |-  O  =  dom  ( id_ `  T
)
3 homib.2 . . . 4  |-  J  =  ( id_ `  T
)
41, 2, 3jdmo 25778 . . 3  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  dom  ( dom_ `  T )
)
5 homib.3 . . . . 5  |-  H  =  ( hom `  T
)
6 eqid 2283 . . . . 5  |-  ( dom_ `  T )  =  (
dom_ `  T )
7 eqid 2283 . . . . 5  |-  ( cod_ `  T )  =  (
cod_ `  T )
81, 5, 6, 7mrdmcd 25794 . . . 4  |-  ( T  e.  Cat OLD  ->  ( ( J `  A
)  e.  dom  ( dom_ `  T )  -> 
( J `  A
)  e.  ( H `
 <. ( ( dom_ `  T ) `  ( J `  A )
) ,  ( (
cod_ `  T ) `  ( J `  A
) ) >. )
) )
98adantr 451 . . 3  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( J `
 A )  e. 
dom  ( dom_ `  T
)  ->  ( J `  A )  e.  ( H `  <. (
( dom_ `  T ) `  ( J `  A
) ) ,  ( ( cod_ `  T
) `  ( J `  A ) ) >.
) ) )
104, 9mpd 14 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  ( H `  <. (
( dom_ `  T ) `  ( J `  A
) ) ,  ( ( cod_ `  T
) `  ( J `  A ) ) >.
) )
11 eqid 2283 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
122, 6, 11, 7idosc 25769 . . . . 5  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( (
dom_ `  T ) `  ( ( id_ `  T
) `  A )
)  =  A  /\  ( ( cod_ `  T
) `  ( ( id_ `  T ) `  A ) )  =  A ) )
133eqcomi 2287 . . . . . . . . . 10  |-  ( id_ `  T )  =  J
1413fveq1i 5526 . . . . . . . . 9  |-  ( ( id_ `  T ) `
 A )  =  ( J `  A
)
1514fveq2i 5528 . . . . . . . 8  |-  ( (
dom_ `  T ) `  ( ( id_ `  T
) `  A )
)  =  ( (
dom_ `  T ) `  ( J `  A
) )
1615eqeq1i 2290 . . . . . . 7  |-  ( ( ( dom_ `  T
) `  ( ( id_ `  T ) `  A ) )  =  A  <->  ( ( dom_ `  T ) `  ( J `  A )
)  =  A )
1716biimpi 186 . . . . . 6  |-  ( ( ( dom_ `  T
) `  ( ( id_ `  T ) `  A ) )  =  A  ->  ( ( dom_ `  T ) `  ( J `  A ) )  =  A )
1817adantr 451 . . . . 5  |-  ( ( ( ( dom_ `  T
) `  ( ( id_ `  T ) `  A ) )  =  A  /\  ( (
cod_ `  T ) `  ( ( id_ `  T
) `  A )
)  =  A )  ->  ( ( dom_ `  T ) `  ( J `  A )
)  =  A )
1912, 18syl 15 . . . 4  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( dom_ `  T ) `  ( J `  A )
)  =  A )
20 catded 25764 . . . . . 6  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
2113dmeqi 4880 . . . . . . . . 9  |-  dom  ( id_ `  T )  =  dom  J
222, 21eqtri 2303 . . . . . . . 8  |-  O  =  dom  J
2322, 6, 3, 7idosd 25744 . . . . . . 7  |-  ( ( T  e.  Ded  /\  A  e.  O )  ->  ( ( ( dom_ `  T ) `  ( J `  A )
)  =  A  /\  ( ( cod_ `  T
) `  ( J `  A ) )  =  A ) )
2423ancomd 438 . . . . . 6  |-  ( ( T  e.  Ded  /\  A  e.  O )  ->  ( ( ( cod_ `  T ) `  ( J `  A )
)  =  A  /\  ( ( dom_ `  T
) `  ( J `  A ) )  =  A ) )
2520, 24sylan 457 . . . . 5  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( (
cod_ `  T ) `  ( J `  A
) )  =  A  /\  ( ( dom_ `  T ) `  ( J `  A )
)  =  A ) )
2625simpld 445 . . . 4  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( ( cod_ `  T ) `  ( J `  A )
)  =  A )
2719, 26opeq12d 3804 . . 3  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  <. ( ( dom_ `  T ) `  ( J `  A )
) ,  ( (
cod_ `  T ) `  ( J `  A
) ) >.  =  <. A ,  A >. )
2827fveq2d 5529 . 2  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( H `  <. ( ( dom_ `  T
) `  ( J `  A ) ) ,  ( ( cod_ `  T
) `  ( J `  A ) ) >.
)  =  ( H `
 <. A ,  A >. ) )
2910, 28eleqtrd 2359 1  |-  ( ( T  e.  Cat OLD  /\  A  e.  O )  ->  ( J `  A )  e.  ( H `  <. A ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   dom cdm 4689   ` cfv 5255   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   Dedcded 25734    Cat
OLD ccatOLD 25752   homchomOLD 25785
This theorem is referenced by:  hine  25797  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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  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  df-homOLD 25786
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