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Theorem homdmcoa 14222
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 2436 . . . 4  |-  (Nat `  C )  =  (Nat
`  C )
2 homdmcoa.h . . . 4  |-  H  =  (Homa
`  C )
31, 2homarw 14201 . . 3  |-  ( X H Y )  C_  (Nat `  C )
4 homdmcoa.f . . 3  |-  ( ph  ->  F  e.  ( X H Y ) )
53, 4sseldi 3346 . 2  |-  ( ph  ->  F  e.  (Nat `  C ) )
61, 2homarw 14201 . . 3  |-  ( Y H Z )  C_  (Nat `  C )
7 homdmcoa.g . . 3  |-  ( ph  ->  G  e.  ( Y H Z ) )
86, 7sseldi 3346 . 2  |-  ( ph  ->  G  e.  (Nat `  C ) )
92homacd 14196 . . . 4  |-  ( F  e.  ( X H Y )  ->  (coda `  F
)  =  Y )
104, 9syl 16 . . 3  |-  ( ph  ->  (coda
`  F )  =  Y )
112homadm 14195 . . . 4  |-  ( G  e.  ( Y H Z )  ->  (domA `  G )  =  Y )
127, 11syl 16 . . 3  |-  ( ph  ->  (domA `  G )  =  Y )
1310, 12eqtr4d 2471 . 2  |-  ( ph  ->  (coda
`  F )  =  (domA `  G ) )
14 homdmcoa.o . . 3  |-  .x.  =  (compa `  C )
1514, 1eldmcoa 14220 . 2  |-  ( G dom  .x.  F  <->  ( F  e.  (Nat `  C )  /\  G  e.  (Nat `  C )  /\  (coda `  F
)  =  (domA `  G ) ) )
165, 8, 13, 15syl3anbrc 1138 1  |-  ( ph  ->  G dom  .x.  F
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   class class class wbr 4212   dom cdm 4878   ` cfv 5454  (class class class)co 6081  domAcdoma 14175  codaccoda 14176  Natcarw 14177  Homachoma 14178  compaccoa 14209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-doma 14179  df-coda 14180  df-homa 14181  df-arw 14182  df-coa 14211
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