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Theorem cofu2nd 14074
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 14071 . . . 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 5724 . . 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 5734 . . . . 5  |-  ( 1st `  G )  e.  _V
7 fvex 5734 . . . . 5  |-  ( 1st `  F )  e.  _V
86, 7coex 5405 . . . 4  |-  ( ( 1st `  G )  o.  ( 1st `  F
) )  e.  _V
9 fvex 5734 . . . . . 6  |-  ( Base `  C )  e.  _V
101, 9eqeltri 2505 . . . . 5  |-  B  e. 
_V
1110, 10mpt2ex 6417 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  e.  _V
128, 11op2nd 6348 . . 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 2483 . 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 733 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
1514fveq2d 5724 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
16 simprr 734 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
1716fveq2d 5724 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( 1st `  F
) `  y )  =  ( ( 1st `  F ) `  Y
) )
1815, 17oveq12d 6091 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( ( 1st `  F ) `  x
) ( 2nd `  G
) ( ( 1st `  F ) `  y
) )  =  ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) ) )
1914, 16oveq12d 6091 . . 3  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x ( 2nd `  F ) y )  =  ( X ( 2nd `  F ) Y ) )
2018, 19coeq12d 5029 . 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 6098 . . . 4  |-  ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  e.  _V
24 ovex 6098 . . . 4  |-  ( X ( 2nd `  F
) Y )  e. 
_V
2523, 24coex 5405 . . 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 6193 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    o. ccom 4874   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   Basecbs 13461    Func cfunc 14043    o.func ccofu 14045
This theorem is referenced by:  cofu2  14075  cofucl  14077  cofuass  14078  cofull  14123  cofth  14124  catciso  14254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-ixp 7056  df-func 14047  df-cofu 14049
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