MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  homadmcd Structured version   Unicode version

Theorem homadmcd 14199
Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
homahom.h  |-  H  =  (Homa
`  C )
Assertion
Ref Expression
homadmcd  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )

Proof of Theorem homadmcd
StepHypRef Expression
1 homahom.h . . . . 5  |-  H  =  (Homa
`  C )
21homarel 14193 . . . 4  |-  Rel  ( X H Y )
3 1st2nd 6395 . . . 4  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
42, 3mpan 653 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
5 1st2ndbr 6398 . . . . . 6  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
62, 5mpan 653 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
71homa1 14194 . . . . 5  |-  ( ( 1st `  F ) ( X H Y ) ( 2nd `  F
)  ->  ( 1st `  F )  =  <. X ,  Y >. )
86, 7syl 16 . . . 4  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F )  = 
<. X ,  Y >. )
98opeq1d 3992 . . 3  |-  ( F  e.  ( X H Y )  ->  <. ( 1st `  F ) ,  ( 2nd `  F
) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F ) >. )
104, 9eqtrd 2470 . 2  |-  ( F  e.  ( X H Y )  ->  F  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >. )
11 df-ot 3826 . 2  |-  <. X ,  Y ,  ( 2nd `  F ) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >.
1210, 11syl6eqr 2488 1  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   <.cop 3819   <.cotp 3820   class class class wbr 4214   Rel wrel 4885   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350  Homachoma 14180
This theorem is referenced by:  arwdmcd  14209  arwlid  14229  arwrid  14230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-ot 3826  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-1st 6351  df-2nd 6352  df-homa 14183
  Copyright terms: Public domain W3C validator