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Theorem nat1st2nd 13874
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 13785 . . . 4  |-  Rel  ( C  Func  D )
3 natrcl.1 . . . . . . 7  |-  N  =  ( C Nat  D )
43natrcl 13873 . . . . . 6  |-  ( A  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
51, 4syl 15 . . . . 5  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
65simpld 445 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 1st2nd 6208 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
82, 6, 7sylancr 644 . . 3  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
95simprd 449 . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
10 1st2nd 6208 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
112, 9, 10sylancr 644 . . 3  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
128, 11oveq12d 5918 . 2  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
131, 12eleqtrd 2392 1  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   <.cop 3677   Rel wrel 4731   ` cfv 5292  (class class class)co 5900   1stc1st 6162   2ndc2nd 6163    Func cfunc 13777   Nat cnat 13864
This theorem is referenced by:  fuccocl  13887  fuclid  13889  fucrid  13890  fucass  13891  fucsect  13895  invfuc  13897  fucpropd  13900  evlfcllem  14044  evlfcl  14045  curfuncf  14061  yonedalem3a  14097  yonedalem3b  14102  yonedainv  14104  yonffthlem  14105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-ixp 6861  df-func 13781  df-nat 13866
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