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Theorem cofu2nd 15496
Description: Value of the morphism part of the functor composition. (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 ) )
cofu2nd.x  |-  ( ph  ->  X  e.  B )
cofu2nd.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
cofu2nd  |-  ( ph  ->  ( X ( 2nd `  ( G  o.func  F )
) Y )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
)  o.  ( X ( 2nd `  F
) Y ) ) )

Proof of Theorem cofu2nd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofuval.b . . . . 5  |-  B  =  ( Base `  C
)
2 cofuval.f . . . . 5  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
3 cofuval.g . . . . 5  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
41, 2, 3cofuval 15493 . . . 4  |-  ( 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 ) ) ) >. )
54fveq2d 5852 . . 3  |-  ( ph  ->  ( 2nd `  ( G  o.func 
F ) )  =  ( 2nd `  <. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >. ) )
6 fvex 5858 . . . . 5  |-  ( 1st `  G )  e.  _V
7 fvex 5858 . . . . 5  |-  ( 1st `  F )  e.  _V
86, 7coex 6735 . . . 4  |-  ( ( 1st `  G )  o.  ( 1st `  F
) )  e.  _V
9 fvex 5858 . . . . . 6  |-  ( Base `  C )  e.  _V
101, 9eqeltri 2486 . . . . 5  |-  B  e. 
_V
1110, 10mpt2ex 6860 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  e.  _V
128, 11op2nd 6792 . . 3  |-  ( 2nd `  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 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 ) ) )
135, 12syl6eq 2459 . 2  |-  ( ph  ->  ( 2nd `  ( G  o.func 
F ) )  =  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) )
14 simprl 756 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
1514fveq2d 5852 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
16 simprr 758 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
1716fveq2d 5852 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( 1st `  F
) `  y )  =  ( ( 1st `  F ) `  Y
) )
1815, 17oveq12d 6295 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  =  ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) ) )
1914, 16oveq12d 6295 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x ( 2nd `  F ) y )  =  ( X ( 2nd `  F ) Y ) )
2018, 19coeq12d 4987 . 2  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( ( ( 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 ) ) )
21 cofu2nd.x . 2  |-  ( ph  ->  X  e.  B )
22 cofu2nd.y . 2  |-  ( ph  ->  Y  e.  B )
23 ovex 6305 . . . 4  |-  ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  e.  _V
24 ovex 6305 . . . 4  |-  ( X ( 2nd `  F
) Y )  e. 
_V
2523, 24coex 6735 . . 3  |-  ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  o.  ( X ( 2nd `  F
) Y ) )  e.  _V
2625a1i 11 . 2  |-  ( ph  ->  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
)  o.  ( X ( 2nd `  F
) Y ) )  e.  _V )
2713, 20, 21, 22, 26ovmpt2d 6410 1  |-  ( ph  ->  ( X ( 2nd `  ( G  o.func  F )
) Y )  =  ( ( ( ( 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 1405    e. wcel 1842   _Vcvv 3058   <.cop 3977    o. ccom 4826   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6781   2ndc2nd 6782   Basecbs 14839    Func cfunc 15465    o.func ccofu 15467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-map 7458  df-ixp 7507  df-func 15469  df-cofu 15471
This theorem is referenced by:  cofu2  15497  cofucl  15499  cofuass  15500  cofull  15545  cofth  15546  catciso  15708
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