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Theorem nat1st2nd 14148
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 14059 . . . 4  |-  Rel  ( C  Func  D )
3 natrcl.1 . . . . . . 7  |-  N  =  ( C Nat  D )
43natrcl 14147 . . . . . 6  |-  ( A  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
51, 4syl 16 . . . . 5  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
65simpld 446 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 1st2nd 6393 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
82, 6, 7sylancr 645 . . 3  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
95simprd 450 . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
10 1st2nd 6393 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
112, 9, 10sylancr 645 . . 3  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
128, 11oveq12d 6099 . 2  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
131, 12eleqtrd 2512 1  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817   Rel wrel 4883   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348    Func cfunc 14051   Nat cnat 14138
This theorem is referenced by:  fuccocl  14161  fuclid  14163  fucrid  14164  fucass  14165  fucsect  14169  invfuc  14171  fucpropd  14174  evlfcllem  14318  evlfcl  14319  curfuncf  14335  yonedalem3a  14371  yonedalem3b  14376  yonedainv  14378  yonffthlem  14379
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-ixp 7064  df-func 14055  df-nat 14140
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