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Theorem homadmcd 15645
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 15639 . . . 4  |-  Rel  ( X H Y )
3 1st2nd 6830 . . . 4  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
42, 3mpan 668 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
5 1st2ndbr 6833 . . . . . 6  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
62, 5mpan 668 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
71homa1 15640 . . . . 5  |-  ( ( 1st `  F ) ( X H Y ) ( 2nd `  F
)  ->  ( 1st `  F )  =  <. X ,  Y >. )
86, 7syl 17 . . . 4  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F )  = 
<. X ,  Y >. )
98opeq1d 4165 . . 3  |-  ( F  e.  ( X H Y )  ->  <. ( 1st `  F ) ,  ( 2nd `  F
) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F ) >. )
104, 9eqtrd 2443 . 2  |-  ( F  e.  ( X H Y )  ->  F  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >. )
11 df-ot 3981 . 2  |-  <. X ,  Y ,  ( 2nd `  F ) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >.
1210, 11syl6eqr 2461 1  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   <.cop 3978   <.cotp 3980   class class class wbr 4395   Rel wrel 4828   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783  Homachoma 15626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-ot 3981  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-1st 6784  df-2nd 6785  df-homa 15629
This theorem is referenced by:  arwdmcd  15655  arwlid  15675  arwrid  15676
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