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Theorem natixp 14139
Description: A natural transformation is a function from the objects of 
C to homomorphisms from  F ( x ) to  G ( x ). (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
)
Assertion
Ref Expression
natixp  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
Distinct variable groups:    x, A    x, F    x, G    x, C    x, K    ph, x    x, D    x, L    x, B    x, J
Allowed substitution hint:    N( x)

Proof of Theorem natixp
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 natixp.2 . . 3  |-  ( ph  ->  A  e.  ( <. F ,  G >. N
<. K ,  L >. ) )
2 natrcl.1 . . . 4  |-  N  =  ( C Nat  D )
3 natixp.b . . . 4  |-  B  =  ( Base `  C
)
4 eqid 2435 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
5 natixp.j . . . 4  |-  J  =  (  Hom  `  D
)
6 eqid 2435 . . . 4  |-  (comp `  D )  =  (comp `  D )
72natrcl 14137 . . . . . . 7  |-  ( A  e.  ( <. F ,  G >. N <. K ,  L >. )  ->  ( <. F ,  G >.  e.  ( C  Func  D
)  /\  <. K ,  L >.  e.  ( C 
Func  D ) ) )
81, 7syl 16 . . . . . 6  |-  ( ph  ->  ( <. F ,  G >.  e.  ( C  Func  D )  /\  <. K ,  L >.  e.  ( C 
Func  D ) ) )
98simpld 446 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
10 df-br 4205 . . . . 5  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylibr 204 . . . 4  |-  ( ph  ->  F ( C  Func  D ) G )
128simprd 450 . . . . 5  |-  ( ph  -> 
<. K ,  L >.  e.  ( C  Func  D
) )
13 df-br 4205 . . . . 5  |-  ( K ( C  Func  D
) L  <->  <. K ,  L >.  e.  ( C 
Func  D ) )
1412, 13sylibr 204 . . . 4  |-  ( ph  ->  K ( C  Func  D ) L )
152, 3, 4, 5, 6, 11, 14isnat 14134 . . 3  |-  ( ph  ->  ( A  e.  (
<. F ,  G >. N
<. K ,  L >. )  <-> 
( A  e.  X_ x  e.  B  (
( F `  x
) J ( K `
 x ) )  /\  A. x  e.  B  A. y  e.  B  A. z  e.  ( x (  Hom  `  C ) y ) ( ( A `  y ) ( <.
( F `  x
) ,  ( F `
 y ) >.
(comp `  D )
( K `  y
) ) ( ( x G y ) `
 z ) )  =  ( ( ( x L y ) `
 z ) (
<. ( F `  x
) ,  ( K `
 x ) >.
(comp `  D )
( K `  y
) ) ( A `
 x ) ) ) ) )
161, 15mpbid 202 . 2  |-  ( ph  ->  ( A  e.  X_ x  e.  B  (
( F `  x
) J ( K `
 x ) )  /\  A. x  e.  B  A. y  e.  B  A. z  e.  ( x (  Hom  `  C ) y ) ( ( A `  y ) ( <.
( F `  x
) ,  ( F `
 y ) >.
(comp `  D )
( K `  y
) ) ( ( x G y ) `
 z ) )  =  ( ( ( x L y ) `
 z ) (
<. ( F `  x
) ,  ( K `
 x ) >.
(comp `  D )
( K `  y
) ) ( A `
 x ) ) ) )
1716simpld 446 1  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   X_cixp 7055   Basecbs 13459    Hom chom 13530  compcco 13531    Func cfunc 14041   Nat cnat 14128
This theorem is referenced by:  natcl  14140  natfn  14141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-ixp 7056  df-func 14045  df-nat 14130
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