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Theorem homdmcoa 13899
Description: If  F : X --> Y and  G : Y
--> Z, then  G and  F are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o  |-  .x.  =  (compa `  C )
homdmcoa.h  |-  H  =  (Homa
`  C )
homdmcoa.f  |-  ( ph  ->  F  e.  ( X H Y ) )
homdmcoa.g  |-  ( ph  ->  G  e.  ( Y H Z ) )
Assertion
Ref Expression
homdmcoa  |-  ( ph  ->  G dom  .x.  F
)

Proof of Theorem homdmcoa
StepHypRef Expression
1 eqid 2283 . . . 4  |-  (Nat `  C )  =  (Nat
`  C )
2 homdmcoa.h . . . 4  |-  H  =  (Homa
`  C )
31, 2homarw 13878 . . 3  |-  ( X H Y )  C_  (Nat `  C )
4 homdmcoa.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
53, 4sseldi 3178 . 2  |-  ( ph  ->  F  e.  (Nat `  C ) )
61, 2homarw 13878 . . 3  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . 3  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3178 . 2  |-  ( ph  ->  G  e.  (Nat `  C ) )
92homacd 13873 . . . 4  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
104, 9syl 15 . . 3  |-  ( ph  ->  (coda
`  F )  =  Y )
112homadm 13872 . . . 4  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
127, 11syl 15 . . 3  |-  ( ph  ->  (domA `  G )  =  Y )
1310, 12eqtr4d 2318 . 2  |-  ( ph  ->  (coda
`  F )  =  (domA `  G ) )
14 homdmcoa.o . . 3  |-  .x.  =  (compa `  C )
1514, 1eldmcoa 13897 . 2  |-  ( G dom  .x.  F  <->  ( F  e.  (Nat `  C )  /\  G  e.  (Nat `  C )  /\  (coda `  F
)  =  (domA `  G ) ) )
165, 8, 13, 15syl3anbrc 1136 1  |-  ( ph  ->  G dom  .x.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858  domAcdoma 13852  codaccoda 13853  Natcarw 13854  Homachoma 13855  compaccoa 13886
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-ot 3650  df-uni 3828  df-iun 3907  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-doma 13856  df-coda 13857  df-homa 13858  df-arw 13859  df-coa 13888
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