MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coafval Structured version   Unicode version

Theorem coafval 14219
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 5728 . . . . . 6  |-  ( c  =  C  ->  (Nat `  c )  =  (Nat
`  C ) )
3 coafval.a . . . . . 6  |-  A  =  (Nat `  C )
42, 3syl6eqr 2486 . . . . 5  |-  ( c  =  C  ->  (Nat `  c )  =  A )
5 biidd 229 . . . . . 6  |-  ( c  =  C  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  h )  =  (domA `  g ) ) )
64, 5rabeqbidv 2951 . . . . 5  |-  ( c  =  C  ->  { h  e.  (Nat `  c )  |  (coda
`  h )  =  (domA `  g ) }  =  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
7 fveq2 5728 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
8 coafval.x . . . . . . . . 9  |-  .xb  =  (comp `  C )
97, 8syl6eqr 2486 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .xb  )
109oveqd 6098 . . . . . . 7  |-  ( c  =  C  ->  ( <. (domA `  f ) ,  (domA `  g
) >. (comp `  c
) (coda
`  g ) )  =  ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) )
1110oveqd 6098 . . . . . 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 3998 . . . . 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 6136 . . . 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 14211 . . . 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 5742 . . . . . 6  |-  (Nat `  C )  e.  _V
163, 15eqeltri 2506 . . . . 5  |-  A  e. 
_V
1716rabex 4354 . . . . 5  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  e.  _V
1816, 17mpt2ex 6425 . . . 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 5806 . . 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 5366 . . . . . . 7  |-  dom compa  C_  Cat
2120sseli 3344 . . . . . 6  |-  ( C  e.  dom compa  ->  C  e.  Cat )
2221con3i 129 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom compa )
23 ndmfv 5755 . . . . 5  |-  ( -.  C  e.  dom compa  ->  (compa `  C )  =  (/) )
2422, 23syl 16 . . . 4  |-  ( -.  C  e.  Cat  ->  (compa `  C )  =  (/) )
253arwrcl 14199 . . . . . . . 8  |-  ( f  e.  A  ->  C  e.  Cat )
2625con3i 129 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  -.  f  e.  A )
2726eq0rdv 3662 . . . . . 6  |-  ( -.  C  e.  Cat  ->  A  =  (/) )
28 eqidd 2437 . . . . . 6  |-  ( -.  C  e.  Cat  ->  { h  e.  A  | 
(coda `  h )  =  (domA `  g
) }  =  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )
29 eqidd 2437 . . . . . 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 6136 . . . . 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 6427 . . . . 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 2484 . . . 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 2471 . . 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 158 . 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 2456 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 set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956   (/)c0 3628   <.cop 3817   <.cotp 3818   dom cdm 4878   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   2ndc2nd 6348  compcco 13541   Catccat 13889  domAcdoma 14175  codaccoda 14176  Natcarw 14177  compaccoa 14209
This theorem is referenced by:  eldmcoa  14220  dmcoass  14221  coaval  14223  coapm  14226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-arw 14182  df-coa 14211
  Copyright terms: Public domain W3C validator