MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofu2 Structured version   Unicode version

Theorem cofu2 14916
Description: Value of the morphism part of the functor composition. (Contributed by Mario Carneiro, 28-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 )
cofu2.h  |-  H  =  ( Hom  `  C
)
cofu2.y  |-  ( ph  ->  R  e.  ( X H Y ) )
Assertion
Ref Expression
cofu2  |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func 
F ) ) Y ) `  R )  =  ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) ) `  (
( X ( 2nd `  F ) Y ) `
 R ) ) )

Proof of Theorem cofu2
StepHypRef Expression
1 cofuval.b . . . 4  |-  B  =  ( Base `  C
)
2 cofuval.f . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
3 cofuval.g . . . 4  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
4 cofu2nd.x . . . 4  |-  ( ph  ->  X  e.  B )
5 cofu2nd.y . . . 4  |-  ( ph  ->  Y  e.  B )
61, 2, 3, 4, 5cofu2nd 14915 . . 3  |-  ( ph  ->  ( X ( 2nd `  ( G  o.func  F )
) Y )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
)  o.  ( X ( 2nd `  F
) Y ) ) )
76fveq1d 5802 . 2  |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func 
F ) ) Y ) `  R )  =  ( ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  o.  ( X ( 2nd `  F
) Y ) ) `
 R ) )
8 cofu2.h . . . 4  |-  H  =  ( Hom  `  C
)
9 eqid 2454 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
10 relfunc 14892 . . . . 5  |-  Rel  ( C  Func  D )
11 1st2ndbr 6734 . . . . 5  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 2, 11sylancr 663 . . . 4  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
131, 8, 9, 12, 4, 5funcf2 14898 . . 3  |-  ( ph  ->  ( X ( 2nd `  F ) Y ) : ( X H Y ) --> ( ( ( 1st `  F
) `  X )
( Hom  `  D ) ( ( 1st `  F
) `  Y )
) )
14 cofu2.y . . 3  |-  ( ph  ->  R  e.  ( X H Y ) )
15 fvco3 5878 . . 3  |-  ( ( ( X ( 2nd `  F ) Y ) : ( X H Y ) --> ( ( ( 1st `  F
) `  X )
( Hom  `  D ) ( ( 1st `  F
) `  Y )
)  /\  R  e.  ( X H Y ) )  ->  ( (
( ( ( 1st `  F ) `  X
) ( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  o.  ( X ( 2nd `  F
) Y ) ) `
 R )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
) `  ( ( X ( 2nd `  F
) Y ) `  R ) ) )
1613, 14, 15syl2anc 661 . 2  |-  ( ph  ->  ( ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) )  o.  ( X ( 2nd `  F
) Y ) ) `
 R )  =  ( ( ( ( 1st `  F ) `
 X ) ( 2nd `  G ) ( ( 1st `  F
) `  Y )
) `  ( ( X ( 2nd `  F
) Y ) `  R ) ) )
177, 16eqtrd 2495 1  |-  ( ph  ->  ( ( X ( 2nd `  ( G  o.func 
F ) ) Y ) `  R )  =  ( ( ( ( 1st `  F
) `  X )
( 2nd `  G
) ( ( 1st `  F ) `  Y
) ) `  (
( X ( 2nd `  F ) Y ) `
 R ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   class class class wbr 4401    o. ccom 4953   Rel wrel 4954   -->wf 5523   ` cfv 5527  (class class class)co 6201   1stc1st 6686   2ndc2nd 6687   Basecbs 14293   Hom chom 14369    Func cfunc 14884    o.func ccofu 14886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-ixp 7375  df-func 14888  df-cofu 14890
This theorem is referenced by:  cofucl  14918  1st2ndprf  15136  uncf2  15167  yonedalem22  15208
  Copyright terms: Public domain W3C validator