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Theorem cofuval 15786
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 15764 . . 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 762 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
43fveq2d 5885 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  g
)  =  ( 1st `  G ) )
5 simprr 764 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
65fveq2d 5885 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
74, 6coeq12d 5018 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  g
)  o.  ( 1st `  f ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
85fveq2d 5885 . . . . . . . 8  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
98dmeqd 5056 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  dom  ( 2nd `  F ) )
10 cofuval.b . . . . . . . . . 10  |-  B  =  ( Base `  C
)
11 relfunc 15766 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
12 cofuval.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
13 1st2ndbr 6856 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1411, 12, 13sylancr 667 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1510, 14funcfn2 15773 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( B  X.  B ) )
16 fndm 5693 . . . . . . . . 9  |-  ( ( 2nd `  F )  Fn  ( B  X.  B )  ->  dom  ( 2nd `  F )  =  ( B  X.  B ) )
1715, 16syl 17 . . . . . . . 8  |-  ( ph  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
1817adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
199, 18eqtrd 2463 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  ( B  X.  B ) )
2019dmeqd 5056 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  dom  ( B  X.  B ) )
21 dmxpid 5073 . . . . 5  |-  dom  ( B  X.  B )  =  B
2220, 21syl6eq 2479 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  B )
233fveq2d 5885 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
246fveq1d 5883 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  x
) )
256fveq1d 5883 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  y )  =  ( ( 1st `  F ) `  y
) )
2623, 24, 25oveq123d 6326 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( 1st `  f ) `  x
) ( 2nd `  g
) ( ( 1st `  f ) `  y
) )  =  ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) ) )
278oveqd 6322 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x ( 2nd `  f ) y )  =  ( x ( 2nd `  F ) y ) )
2826, 27coeq12d 5018 . . . 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 6367 . . 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 4195 . 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 3089 . . 3  |-  ( G  e.  ( D  Func  E )  ->  G  e.  _V )
3331, 32syl 17 . 2  |-  ( ph  ->  G  e.  _V )
34 elex 3089 . . 3  |-  ( F  e.  ( C  Func  D )  ->  F  e.  _V )
3512, 34syl 17 . 2  |-  ( ph  ->  F  e.  _V )
36 opex 4685 . . 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 6438 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 370    = wceq 1437    e. wcel 1872   _Vcvv 3080   <.cop 4004   class class class wbr 4423    X. cxp 4851   dom cdm 4853    o. ccom 4857   Rel wrel 4858    Fn wfn 5596   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806   Basecbs 15120    Func cfunc 15758    o.func ccofu 15760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-map 7485  df-ixp 7534  df-func 15762  df-cofu 15764
This theorem is referenced by:  cofu1st  15787  cofu2nd  15789  cofuval2  15791  cofucl  15792  cofuass  15793  cofulid  15794  cofurid  15795  prf1st  16088  prf2nd  16089
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