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Theorem nat1st2nd 15439
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 15350 . . . 4  |-  Rel  ( C  Func  D )
3 natrcl.1 . . . . . . 7  |-  N  =  ( C Nat  D )
43natrcl 15438 . . . . . 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 457 . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 1st2nd 6819 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
82, 6, 7sylancr 661 . . 3  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
95simprd 461 . . . 4  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
10 1st2nd 6819 . . . 4  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >. )
112, 9, 10sylancr 661 . . 3  |-  ( ph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
128, 11oveq12d 6288 . 2  |-  ( ph  ->  ( F N G )  =  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
131, 12eleqtrd 2544 1  |-  ( ph  ->  A  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022   Rel wrel 4993   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772    Func cfunc 15342   Nat cnat 15429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-ixp 7463  df-func 15346  df-nat 15431
This theorem is referenced by:  fuccocl  15452  fuclid  15454  fucrid  15455  fucass  15456  fucsect  15460  invfuc  15462  fucpropd  15465  evlfcllem  15689  evlfcl  15690  curfuncf  15706  yonedalem3a  15742  yonedalem3b  15747  yonedainv  15749  yonffthlem  15750
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