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Theorem coafval 13912
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 5541 . . . . . 6  |-  ( c  =  C  ->  (Nat `  c )  =  (Nat
`  C ) )
3 coafval.a . . . . . 6  |-  A  =  (Nat `  C )
42, 3syl6eqr 2346 . . . . 5  |-  ( c  =  C  ->  (Nat `  c )  =  A )
5 biidd 228 . . . . . 6  |-  ( c  =  C  ->  (
(coda `  h )  =  (domA `  g
)  <->  (coda
`  h )  =  (domA `  g ) ) )
64, 5rabeqbidv 2796 . . . . 5  |-  ( c  =  C  ->  { h  e.  (Nat `  c )  |  (coda
`  h )  =  (domA `  g ) }  =  { h  e.  A  |  (coda
`  h )  =  (domA `  g ) } )
7 fveq2 5541 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
8 coafval.x . . . . . . . . 9  |-  .xb  =  (comp `  C )
97, 8syl6eqr 2346 . . . . . . . 8  |-  ( c  =  C  ->  (comp `  c )  =  .xb  )
109oveqd 5891 . . . . . . 7  |-  ( c  =  C  ->  ( <. (domA `  f ) ,  (domA `  g
) >. (comp `  c
) (coda
`  g ) )  =  ( <. (domA `  f ) ,  (domA `  g ) >.  .xb  (coda `  g
) ) )
1110oveqd 5891 . . . . . 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 3826 . . . . 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 5926 . . . 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 13904 . . . 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 5555 . . . . . 6  |-  (Nat `  C )  e.  _V
163, 15eqeltri 2366 . . . . 5  |-  A  e. 
_V
1716rabex 4181 . . . . 5  |-  { h  e.  A  |  (coda `  h
)  =  (domA `  g ) }  e.  _V
1816, 17mpt2ex 6214 . . . 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 5618 . . 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 5185 . . . . . . 7  |-  dom compa  C_  Cat
2120sseli 3189 . . . . . 6  |-  ( C  e.  dom compa  ->  C  e.  Cat )
2221con3i 127 . . . . 5  |-  ( -.  C  e.  Cat  ->  -.  C  e.  dom compa )
23 ndmfv 5568 . . . . 5  |-  ( -.  C  e.  dom compa  ->  (compa `  C )  =  (/) )
2422, 23syl 15 . . . 4  |-  ( -.  C  e.  Cat  ->  (compa `  C )  =  (/) )
253arwrcl 13892 . . . . . . . 8  |-  ( f  e.  A  ->  C  e.  Cat )
2625con3i 127 . . . . . . 7  |-  ( -.  C  e.  Cat  ->  -.  f  e.  A )
2726eq0rdv 3502 . . . . . 6  |-  ( -.  C  e.  Cat  ->  A  =  (/) )
28 eqidd 2297 . . . . . 6  |-  ( -.  C  e.  Cat  ->  { h  e.  A  | 
(coda `  h )  =  (domA `  g
) }  =  {
h  e.  A  | 
(coda `  h )  =  (domA `  g
) } )
29 eqidd 2297 . . . . . 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 5926 . . . . 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 6215 . . . . 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 2344 . . . 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 2331 . . 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 2316 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   (/)c0 3468   <.cop 3656   <.cotp 3657   dom cdm 4705   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   2ndc2nd 6137  compcco 13236   Catccat 13582  domAcdoma 13868  codaccoda 13869  Natcarw 13870  compaccoa 13902
This theorem is referenced by:  eldmcoa  13913  dmcoass  13914  coaval  13916  coapm  13919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-arw 13875  df-coa 13904
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