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Theorem dmcoass 13898
Description: The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o  |-  .x.  =  (compa `  C )
coafval.a  |-  A  =  (Nat `  C )
Assertion
Ref Expression
dmcoass  |-  dom  .x.  C_  ( A  X.  A
)

Proof of Theorem dmcoass
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coafval.o . . . 4  |-  .x.  =  (compa `  C )
2 coafval.a . . . 4  |-  A  =  (Nat `  C )
3 eqid 2283 . . . 4  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 13896 . . 3  |-  .x.  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  C ) (coda `  g
) ) ( 2nd `  f ) ) >.
)
54dmmpt2ssx 6189 . 2  |-  dom  .x.  C_ 
U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
6 snssi 3759 . . . . 5  |-  ( g  e.  A  ->  { g }  C_  A )
7 ssrab2 3258 . . . . 5  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  C_  A
8 xpss12 4792 . . . . 5  |-  ( ( { g }  C_  A  /\  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  C_  A )  ->  ( { g }  X.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) } )  C_  ( A  X.  A ) )
96, 7, 8sylancl 643 . . . 4  |-  ( g  e.  A  ->  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } ) 
C_  ( A  X.  A ) )
109rgen 2608 . . 3  |-  A. g  e.  A  ( {
g }  X.  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )  C_  ( A  X.  A
)
11 iunss 3943 . . 3  |-  ( U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } ) 
C_  ( A  X.  A )  <->  A. g  e.  A  ( {
g }  X.  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )  C_  ( A  X.  A
) )
1210, 11mpbir 200 . 2  |-  U_ g  e.  A  ( {
g }  X.  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )  C_  ( A  X.  A
)
135, 12sstri 3188 1  |-  dom  .x.  C_  ( A  X.  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905    X. cxp 4687   dom cdm 4689   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121  compcco 13220  domAcdoma 13852  codaccoda 13853  Natcarw 13854  compaccoa 13886
This theorem is referenced by:  coapm  13903
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-arw 13859  df-coa 13888
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