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Theorem cofu2nd 15103
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 15100 . . . 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 5863 . . 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 5869 . . . . 5  |-  ( 1st `  G )  e.  _V
7 fvex 5869 . . . . 5  |-  ( 1st `  F )  e.  _V
86, 7coex 6728 . . . 4  |-  ( ( 1st `  G )  o.  ( 1st `  F
) )  e.  _V
9 fvex 5869 . . . . . 6  |-  ( Base `  C )  e.  _V
101, 9eqeltri 2546 . . . . 5  |-  B  e. 
_V
1110, 10mpt2ex 6852 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  e.  _V
128, 11op2nd 6785 . . 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 2519 . 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 755 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
1514fveq2d 5863 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( ( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  X
) )
16 simprr 756 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
1716fveq2d 5863 . . . 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 5160 . 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 6302 . . . 4  |-  ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  e.  _V
24 ovex 6302 . . . 4  |-  ( X ( 2nd `  F
) Y )  e. 
_V
2523, 24coex 6728 . . 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 6407 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 369    = wceq 1374    e. wcel 1762   _Vcvv 3108   <.cop 4028    o. ccom 4998   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6774   2ndc2nd 6775   Basecbs 14481    Func cfunc 15072    o.func ccofu 15074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-map 7414  df-ixp 7462  df-func 15076  df-cofu 15078
This theorem is referenced by:  cofu2  15104  cofucl  15106  cofuass  15107  cofull  15152  cofth  15153  catciso  15283
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