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Theorem natixp 14078
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 2389 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
5 natixp.j . . . 4  |-  J  =  (  Hom  `  D
)
6 eqid 2389 . . . 4  |-  (comp `  D )  =  (comp `  D )
72natrcl 14076 . . . . . . 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 4156 . . . . 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 4156 . . . . 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 14073 . . 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 1649    e. wcel 1717   A.wral 2651   <.cop 3762   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   X_cixp 7001   Basecbs 13398    Hom chom 13469  compcco 13470    Func cfunc 13980   Nat cnat 14067
This theorem is referenced by:  natcl  14079  natfn  14080
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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