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Theorem homadmcd 15218
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 15212 . . . 4  |-  Rel  ( X H Y )
3 1st2nd 6822 . . . 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 6825 . . . . . 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 15213 . . . . 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 4214 . . 3  |-  ( F  e.  ( X H Y )  ->  <. ( 1st `  F ) ,  ( 2nd `  F
) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F ) >. )
104, 9eqtrd 2503 . 2  |-  ( F  e.  ( X H Y )  ->  F  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >. )
11 df-ot 4031 . 2  |-  <. X ,  Y ,  ( 2nd `  F ) >.  =  <. <. X ,  Y >. ,  ( 2nd `  F
) >.
1210, 11syl6eqr 2521 1  |-  ( F  e.  ( X H Y )  ->  F  =  <. X ,  Y ,  ( 2nd `  F
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   <.cop 4028   <.cotp 4030   class class class wbr 4442   Rel wrel 4999   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775  Homachoma 15199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-ot 4031  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-1st 6776  df-2nd 6777  df-homa 15202
This theorem is referenced by:  arwdmcd  15228  arwlid  15248  arwrid  15249
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