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Theorem natixp 13842
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 2296 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
5 natixp.j . . . 4  |-  J  =  (  Hom  `  D
)
6 eqid 2296 . . . 4  |-  (comp `  D )  =  (comp `  D )
72natrcl 13840 . . . . . . 7  |-  ( A  e.  ( <. F ,  G >. N <. K ,  L >. )  ->  ( <. F ,  G >.  e.  ( C  Func  D
)  /\  <. K ,  L >.  e.  ( C 
Func  D ) ) )
81, 7syl 15 . . . . . 6  |-  ( ph  ->  ( <. F ,  G >.  e.  ( C  Func  D )  /\  <. K ,  L >.  e.  ( C 
Func  D ) ) )
98simpld 445 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
10 df-br 4040 . . . . 5  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylibr 203 . . . 4  |-  ( ph  ->  F ( C  Func  D ) G )
128simprd 449 . . . . 5  |-  ( ph  -> 
<. K ,  L >.  e.  ( C  Func  D
) )
13 df-br 4040 . . . . 5  |-  ( K ( C  Func  D
) L  <->  <. K ,  L >.  e.  ( C 
Func  D ) )
1412, 13sylibr 203 . . . 4  |-  ( ph  ->  K ( C  Func  D ) L )
152, 3, 4, 5, 6, 11, 14isnat 13837 . . 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 201 . 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 445 1  |-  ( ph  ->  A  e.  X_ x  e.  B  ( ( F `  x ) J ( K `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   X_cixp 6833   Basecbs 13164    Hom chom 13235  compcco 13236    Func cfunc 13744   Nat cnat 13831
This theorem is referenced by:  natcl  13843  natfn  13844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-ixp 6834  df-func 13748  df-nat 13833
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