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Theorem eldmcoa 14175
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 4173 . 2  |-  ( G dom  .x.  F  <->  <. G ,  F >.  e.  dom  .x.  )
2 otex 4388 . . . . . 6  |-  <. (domA `  f ) ,  (coda
`  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  C ) (coda `  g
) ) ( 2nd `  f ) ) >.  e.  _V
32rgen2w 2734 . . . . 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 2404 . . . . . . 7  |-  (comp `  C )  =  (comp `  C )
74, 5, 6coafval 14174 . . . . . 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 6376 . . . . 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 200 . . . 4  |-  .x.  : U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } ) --> _V
109fdmi 5555 . . 3  |-  dom  .x.  =  U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
1110eleq2i 2468 . 2  |-  ( <. G ,  F >.  e. 
dom  .x.  <->  <. G ,  F >.  e.  U_ g  e.  A  ( { g }  X.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) } ) )
12 fveq2 5687 . . . . . 6  |-  ( g  =  G  ->  (domA `  g )  =  (domA `  G ) )
1312eqeq2d 2415 . . . . 5  |-  ( g  =  G  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  h )  =  (domA `  G ) ) )
1413rabbidv 2908 . . . 4  |-  ( g  =  G  ->  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  =  { h  e.  A  |  (coda `  h
)  =  (domA `  G ) } )
1514opeliunxp2 4972 . . 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 5687 . . . . . 6  |-  ( h  =  F  ->  (coda `  h
)  =  (coda `  F
) )
1716eqeq1d 2412 . . . . 5  |-  ( h  =  F  ->  (
(coda `  h )  =  (domA `  G
)  <->  (coda
`  F )  =  (domA `  G ) ) )
1817elrab 3052 . . . 4  |-  ( F  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  G ) }  <->  ( F  e.  A  /\  (coda `  F
)  =  (domA `  G ) ) )
1918anbi2i 676 . . 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 773 . . . 4  |-  ( ( G  e.  A  /\  ( F  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) )  <-> 
( F  e.  A  /\  ( G  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) ) )
21 3anass 940 . . . 4  |-  ( ( F  e.  A  /\  G  e.  A  /\  (coda `  F )  =  (domA `  G
) )  <->  ( F  e.  A  /\  ( G  e.  A  /\  (coda `  F )  =  (domA `  G
) ) ) )
2220, 21bitr4i 244 . . 3  |-  ( ( G  e.  A  /\  ( F  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) )  <-> 
( F  e.  A  /\  G  e.  A  /\  (coda
`  F )  =  (domA `  G ) ) )
2315, 19, 223bitri 263 . 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 263 1  |-  ( G dom  .x.  F  <->  ( F  e.  A  /\  G  e.  A  /\  (coda `  F
)  =  (domA `  G ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670   _Vcvv 2916   {csn 3774   <.cop 3777   <.cotp 3778   U_ciun 4053   class class class wbr 4172    X. cxp 4835   dom cdm 4837   -->wf 5409   ` cfv 5413  (class class class)co 6040   2ndc2nd 6307  compcco 13496  domAcdoma 14130  codaccoda 14131  Natcarw 14132  compaccoa 14164
This theorem is referenced by:  homdmcoa  14177  coapm  14181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-ot 3784  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-arw 14137  df-coa 14166
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