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Theorem eldmcoa 13913
Description: A pair  <. G ,  F >. is in the domain of the arrow composition, if the domain of  G equals the codomain of  F. (In this case we say  G and  F are composable.) (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o  |-  .x.  =  (compa `  C )
coafval.a  |-  A  =  (Nat `  C )
Assertion
Ref Expression
eldmcoa  |-  ( G dom  .x.  F  <->  ( F  e.  A  /\  G  e.  A  /\  (coda `  F
)  =  (domA `  G ) ) )

Proof of Theorem eldmcoa
Dummy variables  f 
g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 4040 . 2  |-  ( G dom  .x.  F  <->  <. G ,  F >.  e.  dom  .x.  )
2 otex 4254 . . . . . 6  |-  <. (domA `  f ) ,  (coda
`  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  C ) (coda `  g
) ) ( 2nd `  f ) ) >.  e.  _V
32rgen2w 2624 . . . . 5  |-  A. g  e.  A  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
) ) >.  e.  _V
4 coafval.o . . . . . . 7  |-  .x.  =  (compa `  C )
5 coafval.a . . . . . . 7  |-  A  =  (Nat `  C )
6 eqid 2296 . . . . . . 7  |-  (comp `  C )  =  (comp `  C )
74, 5, 6coafval 13912 . . . . . 6  |-  .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 ) ) >.
)
87fmpt2x 6206 . . . . 5  |-  ( A. g  e.  A  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 ) ) >.  e.  _V  <->  .x.  : U_ g  e.  A  ( {
g }  X.  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } ) --> _V )
93, 8mpbi 199 . . . 4  |-  .x.  : U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } ) --> _V
109fdmi 5410 . . 3  |-  dom  .x.  =  U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
1110eleq2i 2360 . 2  |-  ( <. G ,  F >.  e. 
dom  .x.  <->  <. G ,  F >.  e.  U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) } ) )
12 fveq2 5541 . . . . . 6  |-  ( g  =  G  ->  (domA `  g )  =  (domA `  G ) )
1312eqeq2d 2307 . . . . 5  |-  ( g  =  G  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  h )  =  (domA `  G ) ) )
1413rabbidv 2793 . . . 4  |-  ( g  =  G  ->  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  =  { h  e.  A  |  (coda `  h
)  =  (domA `  G ) } )
1514opeliunxp2 4840 . . 3  |-  ( <. G ,  F >.  e. 
U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )  <-> 
( G  e.  A  /\  F  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  G ) } ) )
16 fveq2 5541 . . . . . 6  |-  ( h  =  F  ->  (coda `  h
)  =  (coda `  F
) )
1716eqeq1d 2304 . . . . 5  |-  ( h  =  F  ->  (
(coda `  h )  =  (domA `  G
)  <->  (coda
`  F )  =  (domA `  G ) ) )
1817elrab 2936 . . . 4  |-  ( F  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  G ) }  <->  ( F  e.  A  /\  (coda `  F
)  =  (domA `  G ) ) )
1918anbi2i 675 . . 3  |-  ( ( G  e.  A  /\  F  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  G ) } )  <->  ( G  e.  A  /\  ( F  e.  A  /\  (coda `  F )  =  (domA `  G
) ) ) )
20 an12 772 . . . 4  |-  ( ( G  e.  A  /\  ( F  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) )  <-> 
( F  e.  A  /\  ( G  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) ) )
21 3anass 938 . . . 4  |-  ( ( F  e.  A  /\  G  e.  A  /\  (coda `  F )  =  (domA `  G
) )  <->  ( F  e.  A  /\  ( G  e.  A  /\  (coda `  F )  =  (domA `  G
) ) ) )
2220, 21bitr4i 243 . . 3  |-  ( ( G  e.  A  /\  ( F  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) )  <-> 
( F  e.  A  /\  G  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) )
2315, 19, 223bitri 262 . 2  |-  ( <. G ,  F >.  e. 
U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )  <-> 
( F  e.  A  /\  G  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) )
241, 11, 233bitri 262 1  |-  ( G dom  .x.  F  <->  ( F  e.  A  /\  G  e.  A  /\  (coda `  F
)  =  (domA `  G ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039    X. cxp 4703   dom cdm 4705   -->wf 5267   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137  compcco 13236  domAcdoma 13868  codaccoda 13869  Natcarw 13870  compaccoa 13902
This theorem is referenced by:  homdmcoa  13915  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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