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Theorem natcl 15456
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 15455 . 2  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
6 natcl.1 . 2  |-  ( ph  ->  X  e.  B )
7 fveq2 5803 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
8 fveq2 5803 . . . 4  |-  ( x  =  X  ->  ( K `  x )  =  ( K `  X ) )
97, 8oveq12d 6250 . . 3  |-  ( x  =  X  ->  (
( F `  x
) J ( K `
 x ) )  =  ( ( F `
 X ) J ( K `  X
) ) )
109fvixp 7430 . 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 659 1  |-  ( ph  ->  ( A `  X
)  e.  ( ( F `  X ) J ( K `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   <.cop 3975   ` cfv 5523  (class class class)co 6232   X_cixp 7425   Basecbs 14731   Hom chom 14810   Nat cnat 15444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-ixp 7426  df-func 15361  df-nat 15446
This theorem is referenced by:  fuccocl  15467  fuclid  15469  fucrid  15470  fucass  15471  fucsect  15475  invfuc  15477  fucpropd  15480  evlfcllem  15704  evlfcl  15705  curfuncf  15721  yonedalem3a  15757  yonedalem3b  15762  yonedainv  15764  yonffthlem  15765
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