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Theorem nat1st2nd 15189
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 15100 . . . 4  |-  Rel  ( C  Func  D )
3 natrcl.1 . . . . . . 7  |-  N  =  ( C Nat  D )
43natrcl 15188 . . . . . 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 459 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 1st2nd 6827 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
82, 6, 7sylancr 663 . . 3  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
95simprd 463 . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
10 1st2nd 6827 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
112, 9, 10sylancr 663 . . 3  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
128, 11oveq12d 6295 . 2  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
131, 12eleqtrd 2531 1  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   <.cop 4016   Rel wrel 4990   ` cfv 5574  (class class class)co 6277   1stc1st 6779   2ndc2nd 6780    Func cfunc 15092   Nat cnat 15179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-ixp 7468  df-func 15096  df-nat 15181
This theorem is referenced by:  fuccocl  15202  fuclid  15204  fucrid  15205  fucass  15206  fucsect  15210  invfuc  15212  fucpropd  15215  evlfcllem  15359  evlfcl  15360  curfuncf  15376  yonedalem3a  15412  yonedalem3b  15417  yonedainv  15419  yonffthlem  15420
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