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Theorem dmcoass 15057
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 2454 . . . 4  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 15055 . . 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 6752 . 2  |-  dom  .x.  C_ 
U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
6 iunss 4322 . . 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 4128 . . . 4  |-  ( g  e.  A  ->  { g }  C_  A )
8 ssrab2 3548 . . . 4  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  C_  A
9 xpss12 5056 . . . 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 2904 . 2  |-  U_ g  e.  A  ( {
g }  X.  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )  C_  ( A  X.  A
)
125, 11sstri 3476 1  |-  dom  .x.  C_  ( A  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   {crab 2803    C_ wss 3439   {csn 3988   <.cop 3994   <.cotp 3996   U_ciun 4282    X. cxp 4949   dom cdm 4951   ` cfv 5529  (class class class)co 6203   2ndc2nd 6689  compcco 14373  domAcdoma 15011  codaccoda 15012  Natcarw 15013  compaccoa 15045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-ot 3997  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-arw 15018  df-coa 15047
This theorem is referenced by:  coapm  15062
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