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Theorem natcl 14079
Description: A component of a natural transformation is a morphism. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1  |-  N  =  ( C Nat  D )
natixp.2  |-  ( ph  ->  A  e.  ( <. F ,  G >. N
<. K ,  L >. ) )
natixp.b  |-  B  =  ( Base `  C
)
natixp.j  |-  J  =  (  Hom  `  D
)
natcl.1  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
natcl  |-  ( ph  ->  ( A `  X
)  e.  ( ( F `  X ) J ( K `  X ) ) )

Proof of Theorem natcl
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 natrcl.1 . . 3  |-  N  =  ( C Nat  D )
2 natixp.2 . . 3  |-  ( ph  ->  A  e.  ( <. F ,  G >. N
<. K ,  L >. ) )
3 natixp.b . . 3  |-  B  =  ( Base `  C
)
4 natixp.j . . 3  |-  J  =  (  Hom  `  D
)
51, 2, 3, 4natixp 14078 . 2  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
6 natcl.1 . 2  |-  ( ph  ->  X  e.  B )
7 fveq2 5670 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
8 fveq2 5670 . . . 4  |-  ( x  =  X  ->  ( K `  x )  =  ( K `  X ) )
97, 8oveq12d 6040 . . 3  |-  ( x  =  X  ->  (
( F `  x
) J ( K `
 x ) )  =  ( ( F `
 X ) J ( K `  X
) ) )
109fvixp 7005 . 2  |-  ( ( A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) )  /\  X  e.  B )  ->  ( A `  X
)  e.  ( ( F `  X ) J ( K `  X ) ) )
115, 6, 10syl2anc 643 1  |-  ( ph  ->  ( A `  X
)  e.  ( ( F `  X ) J ( K `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   <.cop 3762   ` cfv 5396  (class class class)co 6022   X_cixp 7001   Basecbs 13398    Hom chom 13469   Nat cnat 14067
This theorem is referenced by:  fuccocl  14090  fuclid  14092  fucrid  14093  fucass  14094  fucsect  14098  invfuc  14100  fucpropd  14103  evlfcllem  14247  evlfcl  14248  curfuncf  14264  yonedalem3a  14300  yonedalem3b  14305  yonedainv  14307  yonffthlem  14308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-ixp 7002  df-func 13984  df-nat 14069
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