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Theorem cofuval 15288
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 15266 . . 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 754 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
43fveq2d 5778 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  g
)  =  ( 1st `  G ) )
5 simprr 755 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
65fveq2d 5778 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
74, 6coeq12d 5080 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  g
)  o.  ( 1st `  f ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
85fveq2d 5778 . . . . . . . 8  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
98dmeqd 5118 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  dom  ( 2nd `  F ) )
10 cofuval.b . . . . . . . . . 10  |-  B  =  ( Base `  C
)
11 relfunc 15268 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
12 cofuval.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
13 1st2ndbr 6748 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1411, 12, 13sylancr 661 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1510, 14funcfn2 15275 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( B  X.  B ) )
16 fndm 5588 . . . . . . . . 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 463 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
199, 18eqtrd 2423 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  ( B  X.  B ) )
2019dmeqd 5118 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  dom  ( B  X.  B ) )
21 dmxpid 5135 . . . . 5  |-  dom  ( B  X.  B )  =  B
2220, 21syl6eq 2439 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  B )
233fveq2d 5778 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
246fveq1d 5776 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  x
) )
256fveq1d 5776 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  y )  =  ( ( 1st `  F ) `  y
) )
2623, 24, 25oveq123d 6217 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( 1st `  f ) `  x
) ( 2nd `  g
) ( ( 1st `  f ) `  y
) )  =  ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) ) )
278oveqd 6213 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x ( 2nd `  f ) y )  =  ( x ( 2nd `  F ) y ) )
2826, 27coeq12d 5080 . . . 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 6258 . . 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 4139 . 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 3043 . . 3  |-  ( G  e.  ( D  Func  E )  ->  G  e.  _V )
3331, 32syl 16 . 2  |-  ( ph  ->  G  e.  _V )
34 elex 3043 . . 3  |-  ( F  e.  ( C  Func  D )  ->  F  e.  _V )
3512, 34syl 16 . 2  |-  ( ph  ->  F  e.  _V )
36 opex 4626 . . 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 6329 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 367    = wceq 1399    e. wcel 1826   _Vcvv 3034   <.cop 3950   class class class wbr 4367    X. cxp 4911   dom cdm 4913    o. ccom 4917   Rel wrel 4918    Fn wfn 5491   ` cfv 5496  (class class class)co 6196    |-> cmpt2 6198   1stc1st 6697   2ndc2nd 6698   Basecbs 14634    Func cfunc 15260    o.func ccofu 15262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-map 7340  df-ixp 7389  df-func 15264  df-cofu 15266
This theorem is referenced by:  cofu1st  15289  cofu2nd  15291  cofuval2  15293  cofucl  15294  cofuass  15295  cofulid  15296  cofurid  15297  prf1st  15590  prf2nd  15591
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