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Theorem cofuval 15105
Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval.b  |-  B  =  ( Base `  C
)
cofuval.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
cofuval.g  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
Assertion
Ref Expression
cofuval  |-  ( ph  ->  ( G  o.func  F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
Distinct variable groups:    x, y, B    x, F, y    x, G, y    ph, x, y
Allowed substitution hints:    C( x, y)    D( x, y)    E( x, y)

Proof of Theorem cofuval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cofu 15083 . . 3  |-  o.func  =  (
g  e.  _V , 
f  e.  _V  |->  <.
( ( 1st `  g
)  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >. )
21a1i 11 . 2  |-  ( ph  ->  o.func  =  ( g  e. 
_V ,  f  e. 
_V  |->  <. ( ( 1st `  g )  o.  ( 1st `  f ) ) ,  ( x  e. 
dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >. ) )
3 simprl 755 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
43fveq2d 5868 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  g
)  =  ( 1st `  G ) )
5 simprr 756 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
65fveq2d 5868 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
74, 6coeq12d 5165 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  g
)  o.  ( 1st `  f ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
85fveq2d 5868 . . . . . . . 8  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
98dmeqd 5203 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  dom  ( 2nd `  F ) )
10 cofuval.b . . . . . . . . . 10  |-  B  =  ( Base `  C
)
11 relfunc 15085 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
12 cofuval.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
13 1st2ndbr 6830 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1411, 12, 13sylancr 663 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1510, 14funcfn2 15092 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( B  X.  B ) )
16 fndm 5678 . . . . . . . . 9  |-  ( ( 2nd `  F )  Fn  ( B  X.  B )  ->  dom  ( 2nd `  F )  =  ( B  X.  B ) )
1715, 16syl 16 . . . . . . . 8  |-  ( ph  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
1817adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
199, 18eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  ( B  X.  B ) )
2019dmeqd 5203 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  dom  ( B  X.  B ) )
21 dmxpid 5220 . . . . 5  |-  dom  ( B  X.  B )  =  B
2220, 21syl6eq 2524 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  B )
233fveq2d 5868 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
246fveq1d 5866 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  x
) )
256fveq1d 5866 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  y )  =  ( ( 1st `  F ) `  y
) )
2623, 24, 25oveq123d 6303 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( 1st `  f ) `  x
) ( 2nd `  g
) ( ( 1st `  f ) `  y
) )  =  ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) ) )
278oveqd 6299 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x ( 2nd `  f ) y )  =  ( x ( 2nd `  F ) y ) )
2826, 27coeq12d 5165 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( ( 1st `  f ) `
 x ) ( 2nd `  g ) ( ( 1st `  f
) `  y )
)  o.  ( x ( 2nd `  f
) y ) )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
2922, 22, 28mpt2eq123dv 6341 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) )
307, 29opeq12d 4221 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( ( 1st `  g
)  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >.  =  <. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >. )
31 cofuval.g . . 3  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
32 elex 3122 . . 3  |-  ( G  e.  ( D  Func  E )  ->  G  e.  _V )
3331, 32syl 16 . 2  |-  ( ph  ->  G  e.  _V )
34 elex 3122 . . 3  |-  ( F  e.  ( C  Func  D )  ->  F  e.  _V )
3512, 34syl 16 . 2  |-  ( ph  ->  F  e.  _V )
36 opex 4711 . . 3  |-  <. (
( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >.  e.  _V
3736a1i 11 . 2  |-  ( ph  -> 
<. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >.  e.  _V )
382, 30, 33, 35, 37ovmpt2d 6412 1  |-  ( ph  ->  ( G  o.func  F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033   class class class wbr 4447    X. cxp 4997   dom cdm 4999    o. ccom 5003   Rel wrel 5004    Fn wfn 5581   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780   Basecbs 14486    Func cfunc 15077    o.func ccofu 15079
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-ixp 7467  df-func 15081  df-cofu 15083
This theorem is referenced by:  cofu1st  15106  cofu2nd  15108  cofuval2  15110  cofucl  15111  cofuass  15112  cofulid  15113  cofurid  15114  prf1st  15327  prf2nd  15328
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