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Theorem homadmcd 14909
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 14903 . . . 4  |-  Rel  ( X H Y )
3 1st2nd 6619 . . . 4  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
42, 3mpan 670 . . 3  |-  ( F  e.  ( X H Y )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
5 1st2ndbr 6622 . . . . . 6  |-  ( ( Rel  ( X H Y )  /\  F  e.  ( X H Y ) )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
62, 5mpan 670 . . . . 5  |-  ( F  e.  ( X H Y )  ->  ( 1st `  F ) ( X H Y ) ( 2nd `  F
) )
71homa1 14904 . . . . 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 4064 . . 3  |-  ( F  e.  ( X H Y )  ->  <. ( 1st `  F ) ,  ( 2nd `  F
) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F ) >. )
104, 9eqtrd 2474 . 2  |-  ( F  e.  ( X H Y )  ->  F  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >. )
11 df-ot 3885 . 2  |-  <. X ,  Y ,  ( 2nd `  F ) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >.
1210, 11syl6eqr 2492 1  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   <.cop 3882   <.cotp 3884   class class class wbr 4291   Rel wrel 4844   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575  Homachoma 14890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-ot 3885  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-1st 6576  df-2nd 6577  df-homa 14893
This theorem is referenced by:  arwdmcd  14919  arwlid  14939  arwrid  14940
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