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Theorem coafval 13896
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 5525 . . . . . 6  |-  ( c  =  C  ->  (Nat `  c )  =  (Nat
`  C ) )
3 coafval.a . . . . . 6  |-  A  =  (Nat `  C )
42, 3syl6eqr 2333 . . . . 5  |-  ( c  =  C  ->  (Nat `  c )  =  A )
5 biidd 228 . . . . . 6  |-  ( c  =  C  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  h )  =  (domA `  g ) ) )
64, 5rabeqbidv 2783 . . . . 5  |-  ( c  =  C  ->  { h  e.  (Nat `  c )  |  (coda
`  h )  =  (domA `  g ) }  =  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
7 fveq2 5525 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
8 coafval.x . . . . . . . . 9  |-  .xb  =  (comp `  C )
97, 8syl6eqr 2333 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .xb  )
109oveqd 5875 . . . . . . 7  |-  ( c  =  C  ->  ( <. (domA `  f ) ,  (domA `  g
) >. (comp `  c
) (coda
`  g ) )  =  ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) )
1110oveqd 5875 . . . . . 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 3810 . . . . 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 5910 . . . 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 13888 . . . 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 5539 . . . . . 6  |-  (Nat `  C )  e.  _V
163, 15eqeltri 2353 . . . . 5  |-  A  e. 
_V
1716rabex 4165 . . . . 5  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  e.  _V
1816, 17mpt2ex 6198 . . . 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 5602 . . 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 5169 . . . . . . 7  |-  dom compa  C_  Cat
2120sseli 3176 . . . . . 6  |-  ( C  e.  dom compa  ->  C  e.  Cat )
2221con3i 127 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom compa )
23 ndmfv 5552 . . . . 5  |-  ( -.  C  e.  dom compa  ->  (compa `  C )  =  (/) )
2422, 23syl 15 . . . 4  |-  ( -.  C  e.  Cat  ->  (compa `  C )  =  (/) )
253arwrcl 13876 . . . . . . . 8  |-  ( f  e.  A  ->  C  e.  Cat )
2625con3i 127 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  -.  f  e.  A )
2726eq0rdv 3489 . . . . . 6  |-  ( -.  C  e.  Cat  ->  A  =  (/) )
28 eqidd 2284 . . . . . 6  |-  ( -.  C  e.  Cat  ->  { h  e.  A  | 
(coda `  h )  =  (domA `  g
) }  =  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )
29 eqidd 2284 . . . . . 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 5910 . . . . 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 6199 . . . . 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 2331 . . . 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 2318 . . 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 156 . 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 2303 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 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   (/)c0 3455   <.cop 3643   <.cotp 3644   dom cdm 4689   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2ndc2nd 6121  compcco 13220   Catccat 13566  domAcdoma 13852  codaccoda 13853  Natcarw 13854  compaccoa 13886
This theorem is referenced by:  eldmcoa  13897  dmcoass  13898  coaval  13900  coapm  13903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-arw 13859  df-coa 13888
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