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Theorem dmcoass 13914
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 2296 . . . 4  |-  (comp `  C )  =  (comp `  C )
41, 2, 3coafval 13912 . . 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 6205 . 2  |-  dom  .x.  C_ 
U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
6 snssi 3775 . . . . 5  |-  ( g  e.  A  ->  { g }  C_  A )
7 ssrab2 3271 . . . . 5  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  C_  A
8 xpss12 4808 . . . . 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 2621 . . 3  |-  A. g  e.  A  ( {
g }  X.  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )  C_  ( A  X.  A
)
11 iunss 3959 . . 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 3201 1  |-  dom  .x.  C_  ( A  X.  A
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560    C_ wss 3165   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921    X. cxp 4703   dom cdm 4705   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137  compcco 13236  domAcdoma 13868  codaccoda 13869  Natcarw 13870  compaccoa 13902
This theorem is referenced by:  coapm  13919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-arw 13875  df-coa 13904
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