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Theorem dmcoass 15472
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 2457 . . . 4  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 15470 . . 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 6864 . 2  |-  dom  .x.  C_ 
U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
6 iunss 4373 . . 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
) )
7 snssi 4176 . . . 4  |-  ( g  e.  A  ->  { g }  C_  A )
8 ssrab2 3581 . . . 4  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  C_  A
9 xpss12 5117 . . . 4  |-  ( ( { 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 ) )
107, 8, 9sylancl 662 . . 3  |-  ( g  e.  A  ->  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } ) 
C_  ( A  X.  A ) )
116, 10mprgbir 2821 . 2  |-  U_ g  e.  A  ( {
g }  X.  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )  C_  ( A  X.  A
)
125, 11sstri 3508 1  |-  dom  .x.  C_  ( A  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    e. wcel 1819   {crab 2811    C_ wss 3471   {csn 4032   <.cop 4038   <.cotp 4040   U_ciun 4332    X. cxp 5006   dom cdm 5008   ` cfv 5594  (class class class)co 6296   2ndc2nd 6798  compcco 14724  domAcdoma 15426  codaccoda 15427  Natcarw 15428  compaccoa 15460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-ot 4041  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-arw 15433  df-coa 15462
This theorem is referenced by:  coapm  15477
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