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Theorem eqidob 25898
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 25867 . . 3  |-  ( C  e.  Cat OLD  ->  C  e.  Ded )
2 eqidob.1 . . . . . 6  |-  O  =  dom  J
3 eqid 2296 . . . . . 6  |-  ( dom_ `  C )  =  (
dom_ `  C )
4 eqidob.2 . . . . . 6  |-  J  =  ( id_ `  C
)
5 eqid 2296 . . . . . 6  |-  ( cod_ `  C )  =  (
cod_ `  C )
62, 3, 4, 5idosd 25847 . . . . 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 25847 . . . . 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 5541 . 2  |-  ( ( J `  A )  =  ( J `  B )  ->  (
( dom_ `  C ) `  ( J `  A
) )  =  ( ( dom_ `  C
) `  ( J `  B ) ) )
13 eqeq12 2308 . . . 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 1632    e. wcel 1696   dom cdm 4705   ` cfv 5271   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   Dedcded 25837    Cat
OLD ccatOLD 25855
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-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-ded 25838  df-catOLD 25856
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