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Theorem coafval 15266
Description: The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o  |-  .x.  =  (compa `  C )
coafval.a  |-  A  =  (Nat `  C )
coafval.x  |-  .xb  =  (comp `  C )
Assertion
Ref Expression
coafval  |-  .x.  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)
Distinct variable groups:    f, g, h, A    C, f, g, h
Allowed substitution hints:    .xb ( f, g, h)    .x. ( f, g, h)

Proof of Theorem coafval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 coafval.o . 2  |-  .x.  =  (compa `  C )
2 fveq2 5872 . . . . . 6  |-  ( c  =  C  ->  (Nat `  c )  =  (Nat
`  C ) )
3 coafval.a . . . . . 6  |-  A  =  (Nat `  C )
42, 3syl6eqr 2526 . . . . 5  |-  ( c  =  C  ->  (Nat `  c )  =  A )
5 biidd 237 . . . . . 6  |-  ( c  =  C  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  h )  =  (domA `  g ) ) )
64, 5rabeqbidv 3113 . . . . 5  |-  ( c  =  C  ->  { h  e.  (Nat `  c )  |  (coda
`  h )  =  (domA `  g ) }  =  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
7 fveq2 5872 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
8 coafval.x . . . . . . . . 9  |-  .xb  =  (comp `  C )
97, 8syl6eqr 2526 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .xb  )
109oveqd 6312 . . . . . . 7  |-  ( c  =  C  ->  ( <. (domA `  f ) ,  (domA `  g
) >. (comp `  c
) (coda
`  g ) )  =  ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) )
1110oveqd 6312 . . . . . 6  |-  ( c  =  C  ->  (
( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  c ) (coda `  g
) ) ( 2nd `  f ) )  =  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) )
1211oteq3d 4233 . . . . 5  |-  ( c  =  C  ->  <. (domA `  f ) ,  (coda
`  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  c ) (coda `  g
) ) ( 2nd `  f ) ) >.  =  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)
134, 6, 12mpt2eq123dv 6354 . . . 4  |-  ( c  =  C  ->  (
g  e.  (Nat `  c ) ,  f  e.  { h  e.  (Nat `  c )  |  (coda
`  h )  =  (domA `  g ) }  |->  <.
(domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  c ) (coda `  g
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
) )
14 df-coa 15258 . . . 4  |- compa  =  ( c  e. 
Cat  |->  ( g  e.  (Nat `  c ) ,  f  e.  { h  e.  (Nat `  c )  |  (coda
`  h )  =  (domA `  g ) }  |->  <.
(domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >. (comp `  c ) (coda `  g
) ) ( 2nd `  f ) ) >.
) )
15 fvex 5882 . . . . . 6  |-  (Nat `  C )  e.  _V
163, 15eqeltri 2551 . . . . 5  |-  A  e. 
_V
1716rabex 4604 . . . . 5  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  e.  _V
1816, 17mpt2ex 6872 . . . 4  |-  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)  e.  _V
1913, 14, 18fvmpt 5957 . . 3  |-  ( C  e.  Cat  ->  (compa `  C
)  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
) )
2014dmmptss 5509 . . . . . . 7  |-  dom compa  C_  Cat
2120sseli 3505 . . . . . 6  |-  ( C  e.  dom compa  ->  C  e.  Cat )
2221con3i 135 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom compa )
23 ndmfv 5896 . . . . 5  |-  ( -.  C  e.  dom compa  ->  (compa `  C )  =  (/) )
2422, 23syl 16 . . . 4  |-  ( -.  C  e.  Cat  ->  (compa `  C )  =  (/) )
253arwrcl 15246 . . . . . . . 8  |-  ( f  e.  A  ->  C  e.  Cat )
2625con3i 135 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  -.  f  e.  A )
2726eq0rdv 3825 . . . . . 6  |-  ( -.  C  e.  Cat  ->  A  =  (/) )
28 eqidd 2468 . . . . . 6  |-  ( -.  C  e.  Cat  ->  { h  e.  A  | 
(coda `  h )  =  (domA `  g
) }  =  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )
29 eqidd 2468 . . . . . 6  |-  ( -.  C  e.  Cat  ->  <.
(domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.  =  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)
3027, 28, 29mpt2eq123dv 6354 . . . . 5  |-  ( -.  C  e.  Cat  ->  ( g  e.  A , 
f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  (/) ,  f  e. 
{ h  e.  A  |  (coda
`  h )  =  (domA `  g ) }  |->  <.
(domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
) )
31 mpt20 6362 . . . . 5  |-  ( g  e.  (/) ,  f  e. 
{ h  e.  A  |  (coda
`  h )  =  (domA `  g ) }  |->  <.
(domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)  =  (/)
3230, 31syl6eq 2524 . . . 4  |-  ( -.  C  e.  Cat  ->  ( g  e.  A , 
f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)  =  (/) )
3324, 32eqtr4d 2511 . . 3  |-  ( -.  C  e.  Cat  ->  (compa `  C )  =  ( g  e.  A , 
f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
) )
3419, 33pm2.61i 164 . 2  |-  (compa `  C
)  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)
351, 34eqtri 2496 1  |-  .x.  =  ( g  e.  A ,  f  e.  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  |->  <. (domA `  f ) ,  (coda `  g ) ,  ( ( 2nd `  g
) ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) ( 2nd `  f ) ) >.
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   (/)c0 3790   <.cop 4039   <.cotp 4041   dom cdm 5005   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   2ndc2nd 6794  compcco 14584   Catccat 14936  domAcdoma 15222  codaccoda 15223  Natcarw 15224  compaccoa 15256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-ot 4042  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-arw 15229  df-coa 15258
This theorem is referenced by:  eldmcoa  15267  dmcoass  15268  coaval  15270  coapm  15273
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