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Theorem natcl 14986
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 14985 . 2  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
6 natcl.1 . 2  |-  ( ph  ->  X  e.  B )
7 fveq2 5802 . . . 4  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
8 fveq2 5802 . . . 4  |-  ( x  =  X  ->  ( K `  x )  =  ( K `  X ) )
97, 8oveq12d 6221 . . 3  |-  ( x  =  X  ->  (
( F `  x
) J ( K `
 x ) )  =  ( ( F `
 X ) J ( K `  X
) ) )
109fvixp 7381 . 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 661 1  |-  ( ph  ->  ( A `  X
)  e.  ( ( F `  X ) J ( K `  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   <.cop 3994   ` cfv 5529  (class class class)co 6203   X_cixp 7376   Basecbs 14296   Hom chom 14372   Nat cnat 14974
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-ixp 7377  df-func 14891  df-nat 14976
This theorem is referenced by:  fuccocl  14997  fuclid  14999  fucrid  15000  fucass  15001  fucsect  15005  invfuc  15007  fucpropd  15010  evlfcllem  15154  evlfcl  15155  curfuncf  15171  yonedalem3a  15207  yonedalem3b  15212  yonedainv  15214  yonffthlem  15215
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