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Theorem eqidob 25795
Description: When the identities are equal, the objects are equal. JFM CAT1 th. 45. (Contributed by FL, 24-Apr-2007.)
Hypotheses
Ref Expression
eqidob.1  |-  O  =  dom  J
eqidob.2  |-  J  =  ( id_ `  C
)
Assertion
Ref Expression
eqidob  |-  ( ( C  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( ( J `  A )  =  ( J `  B )  ->  A  =  B ) )

Proof of Theorem eqidob
StepHypRef Expression
1 catded 25764 . . 3  |-  ( C  e.  Cat OLD  ->  C  e.  Ded )
2 eqidob.1 . . . . . 6  |-  O  =  dom  J
3 eqid 2283 . . . . . 6  |-  ( dom_ `  C )  =  (
dom_ `  C )
4 eqidob.2 . . . . . 6  |-  J  =  ( id_ `  C
)
5 eqid 2283 . . . . . 6  |-  ( cod_ `  C )  =  (
cod_ `  C )
62, 3, 4, 5idosd 25744 . . . . 5  |-  ( ( C  e.  Ded  /\  A  e.  O )  ->  ( ( ( dom_ `  C ) `  ( J `  A )
)  =  A  /\  ( ( cod_ `  C
) `  ( J `  A ) )  =  A ) )
763adant3 975 . . . 4  |-  ( ( C  e.  Ded  /\  A  e.  O  /\  B  e.  O )  ->  ( ( ( dom_ `  C ) `  ( J `  A )
)  =  A  /\  ( ( cod_ `  C
) `  ( J `  A ) )  =  A ) )
82, 3, 4, 5idosd 25744 . . . . 5  |-  ( ( C  e.  Ded  /\  B  e.  O )  ->  ( ( ( dom_ `  C ) `  ( J `  B )
)  =  B  /\  ( ( cod_ `  C
) `  ( J `  B ) )  =  B ) )
983adant2 974 . . . 4  |-  ( ( C  e.  Ded  /\  A  e.  O  /\  B  e.  O )  ->  ( ( ( dom_ `  C ) `  ( J `  B )
)  =  B  /\  ( ( cod_ `  C
) `  ( J `  B ) )  =  B ) )
107, 9jca 518 . . 3  |-  ( ( C  e.  Ded  /\  A  e.  O  /\  B  e.  O )  ->  ( ( ( (
dom_ `  C ) `  ( J `  A
) )  =  A  /\  ( ( cod_ `  C ) `  ( J `  A )
)  =  A )  /\  ( ( (
dom_ `  C ) `  ( J `  B
) )  =  B  /\  ( ( cod_ `  C ) `  ( J `  B )
)  =  B ) ) )
111, 10syl3an1 1215 . 2  |-  ( ( C  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( ( ( (
dom_ `  C ) `  ( J `  A
) )  =  A  /\  ( ( cod_ `  C ) `  ( J `  A )
)  =  A )  /\  ( ( (
dom_ `  C ) `  ( J `  B
) )  =  B  /\  ( ( cod_ `  C ) `  ( J `  B )
)  =  B ) ) )
12 fveq2 5525 . 2  |-  ( ( J `  A )  =  ( J `  B )  ->  (
( dom_ `  C ) `  ( J `  A
) )  =  ( ( dom_ `  C
) `  ( J `  B ) ) )
13 eqeq12 2295 . . . 4  |-  ( ( ( ( dom_ `  C
) `  ( J `  A ) )  =  A  /\  ( (
dom_ `  C ) `  ( J `  B
) )  =  B )  ->  ( (
( dom_ `  C ) `  ( J `  A
) )  =  ( ( dom_ `  C
) `  ( J `  B ) )  <->  A  =  B ) )
1413biimpd 198 . . 3  |-  ( ( ( ( dom_ `  C
) `  ( J `  A ) )  =  A  /\  ( (
dom_ `  C ) `  ( J `  B
) )  =  B )  ->  ( (
( dom_ `  C ) `  ( J `  A
) )  =  ( ( dom_ `  C
) `  ( J `  B ) )  ->  A  =  B )
)
1514ad2ant2r 727 . 2  |-  ( ( ( ( ( dom_ `  C ) `  ( J `  A )
)  =  A  /\  ( ( cod_ `  C
) `  ( J `  A ) )  =  A )  /\  (
( ( dom_ `  C
) `  ( J `  B ) )  =  B  /\  ( (
cod_ `  C ) `  ( J `  B
) )  =  B ) )  ->  (
( ( dom_ `  C
) `  ( J `  A ) )  =  ( ( dom_ `  C
) `  ( J `  B ) )  ->  A  =  B )
)
1611, 12, 15syl2im 34 1  |-  ( ( C  e.  Cat OLD  /\  A  e.  O  /\  B  e.  O )  ->  ( ( J `  A )  =  ( J `  B )  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   dom cdm 4689   ` cfv 5255   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   Dedcded 25734    Cat
OLD ccatOLD 25752
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-dom_ 25717  df-cod_ 25718  df-id_ 25719  df-cmpa 25720  df-ded 25735  df-catOLD 25753
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