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

Theorem idsubidsup 25857
Description: The identity of an an objet of the subcategory equals the identity of the object in the supercategory. (Contributed by FL, 2-Nov-2009.)
Hypotheses
Ref Expression
idsubidsup.1  |-  I1  =  ( id_ `  T )
idsubidsup.2  |-  I 2  =  ( id_ `  U
)
idsubidsup.3  |-  O 2  =  dom  ( id_ `  U
)
Assertion
Ref Expression
idsubidsup  |-  ( U  e.  (  SubCat  `  T
)  ->  A. x  e.  O 2  ( I 2 `  x )  =  ( I1 `  x
) )
Distinct variable groups:    x, T    x, U
Allowed substitution hints:    O 2( x)    I1( x)    I 2( x)

Proof of Theorem idsubidsup
StepHypRef Expression
1 besubbeca 25848 . . . . . 6  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  Cat OLD  )
2 catded 25764 . . . . . . 7  |-  ( T  e.  Cat OLD  ->  T  e.  Ded )
3 dedalg 25743 . . . . . . 7  |-  ( T  e.  Ded  ->  T  e.  Alg  )
42, 3syl 15 . . . . . 6  |-  ( T  e.  Cat OLD  ->  T  e.  Alg  )
51, 4syl 15 . . . . 5  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  Alg  )
6 eqid 2283 . . . . . . 7  |-  dom  ( dom_ `  T )  =  dom  ( dom_ `  T
)
7 eqid 2283 . . . . . . 7  |-  ( dom_ `  T )  =  (
dom_ `  T )
8 eqid 2283 . . . . . . 7  |-  dom  I1  =  dom  I1
9 idsubidsup.1 . . . . . . 7  |-  I1  =  ( id_ `  T )
106, 7, 8, 9ida 25730 . . . . . 6  |-  ( T  e.  Alg  ->  I1 : dom  I1 --> dom  ( dom_ `  T ) )
11 ffun 5391 . . . . . 6  |-  ( I1 : dom  I1 --> dom  ( dom_ `  T )  ->  Fun  I1 )
1210, 11syl 15 . . . . 5  |-  ( T  e.  Alg  ->  Fun  I1 )
135, 12syl 15 . . . 4  |-  ( U  e.  (  SubCat  `  T
)  ->  Fun  I1 )
1413adantr 451 . . 3  |-  ( ( U  e.  (  SubCat  `  T )  /\  x  e.  O 2 )  ->  Fun  I1 )
15 idsubidsup.2 . . . . 5  |-  I 2  =  ( id_ `  U
)
169, 15idsubc 25851 . . . 4  |-  ( U  e.  (  SubCat  `  T
)  ->  I 2  C_  I1 )
1716adantr 451 . . 3  |-  ( ( U  e.  (  SubCat  `  T )  /\  x  e.  O 2 )  ->  I 2  C_  I1 )
18 idsubidsup.3 . . . . . . 7  |-  O 2  =  dom  ( id_ `  U
)
1915eqcomi 2287 . . . . . . . 8  |-  ( id_ `  U )  =  I 2
2019dmeqi 4880 . . . . . . 7  |-  dom  ( id_ `  U )  =  dom  I 2
2118, 20eqtri 2303 . . . . . 6  |-  O 2  =  dom  I 2
2221eleq2i 2347 . . . . 5  |-  ( x  e.  O 2  <->  x  e.  dom  I 2 )
2322biimpi 186 . . . 4  |-  ( x  e.  O 2  ->  x  e.  dom  I 2
)
2423adantl 452 . . 3  |-  ( ( U  e.  (  SubCat  `  T )  /\  x  e.  O 2 )  ->  x  e.  dom  I 2
)
25 funssfv 5543 . . . 4  |-  ( ( Fun  I1  /\  I 2  C_  I1  /\  x  e.  dom  I 2 )  ->  ( I1 `  x )  =  ( I 2 `  x ) )
2625eqcomd 2288 . . 3  |-  ( ( Fun  I1  /\  I 2  C_  I1  /\  x  e.  dom  I 2 )  ->  (
I 2 `  x
)  =  ( I1 `  x ) )
2714, 17, 24, 26syl3anc 1182 . 2  |-  ( ( U  e.  (  SubCat  `  T )  /\  x  e.  O 2 )  -> 
( I 2 `  x )  =  (
I1 `  x )
)
2827ralrimiva 2626 1  |-  ( U  e.  (  SubCat  `  T
)  ->  A. x  e.  O 2  ( I 2 `  x )  =  ( I1 `  x
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   dom cdm 4689   Fun wfun 5249   -->wf 5251   ` cfv 5255    Alg calg 25711   dom_cdom_ 25712   id_cid_ 25714   Dedcded 25734    Cat
OLD ccatOLD 25752    SubCat csubcat 25843
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-pw 3627  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  df-subcat 25844
  Copyright terms: Public domain W3C validator