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Theorem coafval 16037
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 5879 . . . . . 6  |-  ( c  =  C  ->  (Nat `  c )  =  (Nat
`  C ) )
3 coafval.a . . . . . 6  |-  A  =  (Nat `  C )
42, 3syl6eqr 2523 . . . . 5  |-  ( c  =  C  ->  (Nat `  c )  =  A )
54rabeqdv 3025 . . . . 5  |-  ( c  =  C  ->  { h  e.  (Nat `  c )  |  (coda
`  h )  =  (domA `  g ) }  =  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
6 fveq2 5879 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
7 coafval.x . . . . . . . . 9  |-  .xb  =  (comp `  C )
86, 7syl6eqr 2523 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .xb  )
98oveqd 6325 . . . . . . 7  |-  ( c  =  C  ->  ( <. (domA `  f ) ,  (domA `  g
) >. (comp `  c
) (coda
`  g ) )  =  ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) )
109oveqd 6325 . . . . . 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 ) ) )
1110oteq3d 4172 . . . . 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 ) ) >.
)
124, 5, 11mpt2eq123dv 6372 . . . 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 ) ) >.
) )
13 df-coa 16029 . . . 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 ) ) >.
) )
14 fvex 5889 . . . . . 6  |-  (Nat `  C )  e.  _V
153, 14eqeltri 2545 . . . . 5  |-  A  e. 
_V
1615rabex 4550 . . . . 5  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  e.  _V
1715, 16mpt2ex 6889 . . . 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
1812, 13, 17fvmpt 5963 . . 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 ) ) >.
) )
1913dmmptss 5338 . . . . . . 7  |-  dom compa  C_  Cat
2019sseli 3414 . . . . . 6  |-  ( C  e.  dom compa  ->  C  e.  Cat )
2120con3i 142 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom compa )
22 ndmfv 5903 . . . . 5  |-  ( -.  C  e.  dom compa  ->  (compa `  C )  =  (/) )
2321, 22syl 17 . . . 4  |-  ( -.  C  e.  Cat  ->  (compa `  C )  =  (/) )
243arwrcl 16017 . . . . . . . 8  |-  ( f  e.  A  ->  C  e.  Cat )
2524con3i 142 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  -.  f  e.  A )
2625eq0rdv 3773 . . . . . 6  |-  ( -.  C  e.  Cat  ->  A  =  (/) )
27 eqidd 2472 . . . . . 6  |-  ( -.  C  e.  Cat  ->  { h  e.  A  | 
(coda `  h )  =  (domA `  g
) }  =  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )
28 eqidd 2472 . . . . . 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 ) ) >.
)
2926, 27, 28mpt2eq123dv 6372 . . . . 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 ) ) >.
) )
30 mpt20 6380 . . . . 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 ) ) >.
)  =  (/)
3129, 30syl6eq 2521 . . . 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 ) ) >.
)  =  (/) )
3223, 31eqtr4d 2508 . . 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 ) ) >.
) )
3318, 32pm2.61i 169 . 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 ) ) >.
)
341, 33eqtri 2493 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 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   (/)c0 3722   <.cop 3965   <.cotp 3967   dom cdm 4839   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   2ndc2nd 6811  compcco 15280   Catccat 15648  domAcdoma 15993  codaccoda 15994  Natcarw 15995  compaccoa 16027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-ot 3968  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-arw 16000  df-coa 16029
This theorem is referenced by:  eldmcoa  16038  dmcoass  16039  coaval  16041  coapm  16044
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