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Theorem nat1st2nd 15849
Description: Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
natrcl.1  |-  N  =  ( C Nat  D )
nat1st2nd.2  |-  ( ph  ->  A  e.  ( F N G ) )
Assertion
Ref Expression
nat1st2nd  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )

Proof of Theorem nat1st2nd
StepHypRef Expression
1 nat1st2nd.2 . 2  |-  ( ph  ->  A  e.  ( F N G ) )
2 relfunc 15760 . . . 4  |-  Rel  ( C  Func  D )
3 natrcl.1 . . . . . . 7  |-  N  =  ( C Nat  D )
43natrcl 15848 . . . . . 6  |-  ( A  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
51, 4syl 17 . . . . 5  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
65simpld 461 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 1st2nd 6836 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
82, 6, 7sylancr 668 . . 3  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
95simprd 465 . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
10 1st2nd 6836 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
112, 9, 10sylancr 668 . . 3  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
128, 11oveq12d 6306 . 2  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
131, 12eleqtrd 2530 1  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1443    e. wcel 1886   <.cop 3973   Rel wrel 4838   ` cfv 5581  (class class class)co 6288   1stc1st 6788   2ndc2nd 6789    Func cfunc 15752   Nat cnat 15839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-ixp 7520  df-func 15756  df-nat 15841
This theorem is referenced by:  fuccocl  15862  fuclid  15864  fucrid  15865  fucass  15866  fucsect  15870  invfuc  15872  fucpropd  15875  evlfcllem  16099  evlfcl  16100  curfuncf  16116  yonedalem3a  16152  yonedalem3b  16157  yonedainv  16159  yonffthlem  16160
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